The National Gallery will be the first major London gallery to reopen its doors, welcoming visitors from 8 July.
The vast majority of English galleries and museums won't reopen on Saturday, 4 July, despite being given the go-ahead by the government to do so last week.
The Royal Academy also announced its plans on Tuesday, reopening from 9 July, with face masks compulsory. The Barbican gallery will open on 13 July.
Tate said its four venues would not reopen their doors until 27 July.
Many other venues have not yet set a firm date, but some will not let the public back in until August or September.
Last week, the government confirmed that museums and galleries can open from Saturday, as can cinemas, pubs, restaurants, hotels, hairdressers, libraries, theme parks and zoos.
Cultural institutions opening on that date will include Scarborough Art Gallery. But many will not have had time to implement the government safety guidance.
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In their announcements on Tuesday, the National Gallery, Tate, Barbican and Royal Academy all said visitors would need to book timed tickets in advance.
The National Gallery's Titian exhibition, which shut just three days after opening in March, has been extended to 17 January.
Visitors will be asked to follow one-way routes around the building, and to keep to 2m social distancing. The gallery said "higher efficiency filters" have been installed in the air-conditioning system to help the flow of fresh air. Face masks will be "recommended" for visitors.
The Royal Academy will allow members of its Friends scheme back in from 9 July and others a week later, and will only open from Thursdays to Sundays until 2 August. There will also be social distancing in its exhibitions, including Picasso On Paper.
Chief executive Axel Rüger said: "As visitor capacity will be greatly reduced due to social distancing, it will be an opportunity for a quieter, more contemplative experience in the galleries."
The other venues opening soon include the RAF Museum in London on 6 July; and Derby Museum & Art Gallery and the National Army Museum in London, both from 7 July. The capital's Foundling Museum and the Focal Point Gallery in Southend-on-Sea will reopen the following day.
In London, the Whitechapel Gallery and the Photographers' Gallery will open on 14 July, and the Wallace Collection a day later.
Turner Contemporary in Margate, Kent, will follow on 22 July, with The Hepworth Wakefield, Nottingham Contemporary and the Serpentine in early August. But the Whitworth in Manchester is among those that will not let visitors back until September.
The British Museum, Natural History Museum, National Museums Liverpool and Royal Armouries are among those who have not set a reopening date, but have confirmed it won't be 4 July.
A statement from the British Museum said: "Our collection and vast historic building are very complex - and so we're currently planning safe access for visitors and the collection with social distancing."
Venues have been working to put government safety guidelines in place since they were published last Thursday.
They are required to collect a record of visitors for the government's track and trace system, which is a challenge for those sites that do not normally require people to book and give their personal details.
Some council-run venues must also wait until the return of staff who have been seconded to work for frontline services during the pandemic.