In general, wherever you're thinking of going and whatever your situation, it's always worth having a google to see what discounts are available.
Most places are pretty good at putting the information on their own website, which will be the most reliable, but isn't the British Museum that you mention free to all anyway?
As well as the discounts for people on UC, examples I can think of off the top of my head:
Schemes for public sector workers like Boundless, Blue Light card, Edenred etc, so if this applies to you, always worth seeing what you can join - often you pay a fairly token monthly amount, which gives free or discounted entry for lots of places.
I have a National Art pass which gives free entry to museums and galleries, and discounts to exhibitions. Some of these are set in country parks, eg Yorkshire Sculpture Park so has wider appeal than just a 'boring art gallery'.
National Trust membership gives access to grounds, playparks, woods etc where DC can run around, plus free parking, even if you never enter the houses. Likewise English Heritage.
There's schemes for 2 for 1 if you arrive by train, or get vouchers off cereal packets or similar.
In short, it's rare that you have to pay the headline price and there's usually loads of ways to get in for less, but I wouldn't be relying on third party websites as my primary source of information, as it's rare that they're up to date.