The government is committed to maintaining free entry to the permanent collections of national museums, as outlined in the 2019 manifesto.
As culture is a devolved matter, information would not be held by my department on a UK basis. We do not hold complete information on all museums in England. However, we do hold this information for the fifteen DCMS sponsored museum groups, which encompass 40 museums and galleries. Six groups are either partly or wholly based outside of London (National Museums Liverpool, Royal Armouries, Natural History Museum, Tate, Science Museum Group, Imperial War Museums), encompassing half the total number of sponsored museum sites. Visitor numbers for these museum groups are published online: https://www.gov.uk/government/statistics/sponsored-museums-and-galleries-annual-performance-indicators-201819
As the national development agency for museums, Arts Council England collects some information about museums and their visitors. It directly funds a number of regional museums through the National Portfolio and publishes data about their visitors figures. National Portfolio museums (and, pre-2018, Major Partner Museums) have a mix of operating models with some free to enter, some charging, and some charging for special exhibitions. Many charging museums also offer free or concessionary entry to local residents. https://www.artscouncil.org.uk/our-investment/national-portfolio-2018-22
Furthermore, the government supports free entry to regional museums through tax incentives like the VAT 33A relief scheme, and is committed to expanding access to museums across England.