Countrywide museum pass for UK

There's the National Trust and the English Heritage. Is that what you mean?

Yes, that is what I remember now.

National Trust and the English Heritage are not museums. They are both organisations owning old houses, castles etc.
The National Trust (website: https://www.nationaltrust.org.uk/ ) is a venerable British Institution. It is a non-profit making trust that has bought or been donated many stately homes and large areas of countryside for preservation. It charges entrance fees to the houses and gardens, but members get in free (and are not charged for car parks). The areas of land are all open access (though it does charge for car parking - except to members).
English Heritage (website: http://www.english-heritage.org.uk/ ) manages all the government owned historical property. Including Stonehenge.

We got both memberships for each of our trips. For, us (two, month-long trips in 2013 & 2014, England, Wales & Scotland both times), it was worth it.

There used to be a Great British Heritage Pass that could get you into National Trust and English Heritage properties and even a number of privately owned places like Blenheim Palace. That was discontinued because apparently it was too good a deal, and the properties were losing money on it. So it is now best to either join the National Trust or English Heritage. As mentioned there are lots of free museums in London, Oxford and Cambridge.

While there's no National pass, below are links to various city/regional passes, 2 4 1 offers, etc, which might be useful.
London Pass - https://www.londonpass.com/?aid=60&utm_source=bing&utm_medium=cpc&utm_campaign=Brand%7CSearch%7CGBR&utm_term=london%20passes&utm_content=Passes%20London
York Pass - http://www.yorkpass.com/
Days out guide offers 2 4 1 deals throughout the country in conjunction with National Rail - https://www.daysoutguide.co.uk/
2 4 1 entry to British gardens with Gardeners World Magazine - http://www.gardenersworld.com/2-for-1-gardens/order-now/
Special offers for the South West of England via Great Western Trains - https://www.gwr.com/explore-the-west?rd=fgw