The Central Library in Sheffield opened in 1934, when my father was four. As a teenager and medical student in the 1940s and 50s, he would work there, an escape from the confined, cramped home he grew up in. It was, he remembers, "state of the art": a grand public building with elegant, deco curves to the custom-made furniture, handsome timber bookshelves and bright brass handrails up the stairs. On the top floor there was an art gallery, named for JG Graves, who had funded the building – a magnate who made his money from mail-order catalogues.
The library and the Graves Gallery are still there. The latter is part of Museums Sheffield, and has a wonderful collection, including a great Paul Nash, a beautiful Eric Ravilious, a Murillo – and, startlingly, two Italian Renaissance paintings found in the basement of the National Union of Mineworkers' headquarters in Barnsley in the 1960s.
But times are hard. The building remains elegant, but is now rather worn and tatty. Some of the 1930s furniture is still there, but much has been replaced by unprepossessing office desks and chairs. The paint is peeling from the toilet walls. It is worth the visit for the art on the top floor, but pick your time carefully: the gallery is open only four days a week, and shuts at 3pm. In common with other councils, Sheffield is wondering about the purpose of libraries in an information (and austerity) age, and is "consulting" on the future of its service. Elegant architects' plans for the building's refurbishment were drawn up some years ago, but put aside when money got tight. This building, built as an act of philanthropy in the teeth of the Depression, now needs another JG Graves, but, says the museum's director, Kim Streets: "I've yet to meet him."
As the triumphal Olympic and Paralympic summer fades, and time is called on the large-scale, knock-'em-dead Cultural Olympiad work that was made to showcase British arts to a watching world, the industry faces a grim autumn. It is now, after Britain's moment in the sun, that the cuts implemented in the spring will begin to show. The picture is complex: England has lost 30% of its arts council budget, but with the proviso that only 15% should be passed on to "frontline" arts organisations. By March, English local-authority spending on the arts and culture will be down by 16% since 2009-10, but with the picture varying wildly: Somerset County Council, for example, has cut its arts provision by 100%. Scotland, notwithstanding damaging rows between artists and its funding body, Creative Scotland, is on standstill funding.
How are people faring on the ground? This spring, Museums Sheffield's application to become part of Arts Council England's national portfolio was turned down, meaning a loss of £800,000 annual funding. Its income from Sheffield council has also drifted downwards, from £2.38m in 2009-10 to £2.04m in 2012-13. At the same time, energy bills – a significant outgoing for any museum – have been soaring.
Along with the Graves, the museums trust also runs Weston Park Museum, which focuses on social and natural history, and the city-centre Millennium Gallery, devoted to art, craft and design. The cuts have had an undeniably large impact on the programme. "If you'd come here 10 years ago," says exhibitions manager Kirstie Hamilton, "you would have seen 30 shows a year. Now we are hoping to do about 10 – and what would have been a 10-week exhibition is now scheduled to last six months." The cuts provoked local outcry, with hundreds of visitors rallying in support, many registering their outrage on the museum's blog. On one wall of the Graves is a Damien Hirst, lent by Jarvis Cocker to his hometown's museum when he heard about its plight.
Staff are being shed, and fast. From a staff of 108, this autumn they will be 70. By far the hardest-hit department is education. Laura Trevis, recently part of a team of 23, is now one of three. They used to run 24 education workshops a week; that's down to eight. To cater to their 14,000 school visitors a year, she tells me, as she walks me through an exhibition on magic at Weston Park (themed around children's literature, with a Narnia wardrobe and faux-fur coats), they will be offering training to teachers, so that they can guide their own pupils round the exhibits. She is upbeat about the possibilities, but admits: "I think schools may look for other places to take their pupils, at least initially."
When I speak to Stephen Carley, who teaches art at Edward VII School in Sheffield, he is more pessimistic: "It's an absolute tragedy. There's no hiding from that." He mentions the museums' Youth Forum scheme, now gone, which his Year 10s could sign up for and continue with through their A levels: it meant long-term involvement with curators and artists, and hands-on museum experience. "Even with the most inspiring teaching, you can't do that in the classroom. It made the world of art and design real." His ex-pupils number graphic designers, architects – the sort of jobs that are supposed to be fuelling Britain's creative, information-based economy. Many, says Carley, cite that sustained contact with the museum as hugely influential: "That's what the younger pupils are going to miss out on." The arts council has given Museums Sheffield money to tide them over the redundancy period, and they have applied for further funds, but the future remains uncertain.
Museums Sheffield is not the only organisation in the city to have suffered. To the dismay of most who knew their work, Third Angel, a small experimental performance company based in Sheffield, has lost 100% of its Arts Council England grant of just over £33,360 – not even peanuts in the big picture of public spending, but lifeblood to them. At the Edinburgh fringe this year, I saw the company's delicate, thoughtful show What I Heard About the World – a meditation on how the world is increasingly within our grasp, and yet all too frequently experienced via substitutes or simulcra.
It is fantasy to think that Third Angel could exist without public support, and yet it is the kind of work that, eventually, trickles through the system to invigorate the mainstream stages. I meet Alexander Kelly – who, with Rachael Walton, is co-artistic director of Third Angel – one day in Edinburgh. He recalls the morning they got the funding rejection letter, in early 2011. "We weren't all that surprised. We'd been at standstill funding and had been turned down for every other stream of funding we'd applied for at least once, so we knew we weren't a priority." He thinks the arts council "didn't really get what we were doing – but there is a much better understanding of that now". There was an outpouring of protest from fans and, though ACE did not reverse their decision, they did offer some support for the year ahead. Despite the huge setback, packing in wasn't an option: "We've always said that we would carry on as long as we want to make shows together, and we still do," says Kelly. There has been soul-searching and some major changes: Third Angel is now company-in-residence at Sheffield Theatres, sharing ideas and office space, a move that is "properly exciting", says Kelly. Their situation is still precarious: they have no funding in place beyond spring 2013. But, assuming these artists are not defeated in one last, exhausting tussle with a funding application form, the work will go on.
At the opposite end of the country, I arrive in a sun-warmed farmyard at the end of a labyrinth of high-hedged lanes: the office of Take Art, an organisation that brings dance, theatre and performance to audiences in rural Somerset. Its neighbours, in the rose-covered outhouses, are an architect and a blacksmith.
But in the office, things are not as idyllic as they look. The workforce has halved since 2010. Against the wall lie banners with the slogan "We value the arts: against 100% arts cuts", left over from 2010's fruitless struggle against Somerset Council. They have lost around £70,000 a year, taking into account cuts from local councils, too. They still have their arts council funding of £159,000 a year (£30,000 less than they asked for). But the result is that they are providing half the shows they used to. They have launched a fundraising scheme, with hopes to raise £10,000 – though chief executive Ralph Lister is unhappy that Take Art was turned down for ACE's Catalyst scheme, which offers support for fundraising and development. And, though he praises Bristol's Old Vic theatre, with whom Take Art is collaborating, he wishes there were more solidarity shown by the big, national companies. "I've heard a lot of talk about the responsibility of the larger organisations to connect with the smaller ones. Why doesn't [National Theatre director] Nick Hytner call up and say, 'Can we work on a proposal for a rural touring show?'"
One of the organisation's most prominent schemes is its Rural Touring programme. It's a way of getting excellent-quality performances to village halls, and serving areas where access to the arts would otherwise be nonexistent: for 30% of their audiences, says Lister, the work that Take Art brings is the only art they get to see. Often this is quite adventurous: experimental theatre company Kneehigh (Brief Encounter, The Red Shoes) was a regular in their earlier days. Lister points out that, in rural areas, getting in the car and driving to Bath or Bristol may not be an option, as fuel prices rise and pressure on household purses increases; meanwhile theatres in the smaller towns, such as the Merlin in Frome, have been hammered by Somerset's cuts and are clinging on to life.
Take Art offers a "menu" of different shows to village halls, and can subsidise the work to make it affordable – theoretically. In some areas, such as Mendip, the loss of council funding means the subsidy is gone, and the villages can no longer take on the risk. Frances Horler, a retired legal adviser from Kilmersdon near Frome, tells me she has been bringing Take Art shows to her village for 15 years – until now. "We built up an audience," she says. "It's all part of the community feeling of the village." For her, it wasn't just about the evenings out, but a way of gathering people together, and making a few quid to keep the village hall going. Max Miller, from Chilcompton in North Mendip, agrees. "It's a way of people getting to know each other, like the church, or the pub, or post office." Sometimes performers would get involved in workshops at the local primary school. "It's been a strong part of what the village is – its identity." Despite the loss of subsidy, Miller plans to book future shows at full price, but he'll be go for the less adventurous stuff. "In the past, we've had musicians from all over the world – Africa, Eastern Europe – and without the subsidy we wouldn't have been able to take the risk."
The arts are not yet at crisis point. There is no apocalypse, but the damage is real. There will be changes, some of them obvious: a local museum reducing its hours and programme; a local theatre having more dark nights, and fewer shows of real imagination. Other changes will be less tangible: a teenager not getting the spark of inspiration that makes her decide to train as an architect, a village community less bound together by shared experience. Arts organisations are, for the most part, putting a brave face on things, working out how to adjust, not wanting to admit there will be any diminution in what they can do; artists like Alexander Kelly will find ways to make art because that is their life's calling. Walking round the busy, impressive museums in Sheffield, it's clear that those who work here are coping as well as they can. It's still a great place to visit.
But I wonder: how long can you chip away at what a museum does before it starts to look down-at-heel? How far can you cut an exhibition programme before the place feels static? How far can you strip away a staff and still expect them to perform a crucial role in the education of a city's young? What is the point at which a museum, at some quiet, unmarked moment, stops being a questioning, contemplative space at the heart of its city's intellectual and imaginative life, and becomes an irrelevance? The fear is, here and everywhere, that this autumn's hardships are just the beginning: there will come a point when no amount of ingenuity, flexibility and cunning can stand in the way of empty coffers.