What are must see's in England and Scotland while avoiding tourist traps?

My husband is going to the UK for work mid-June. He will be near York. I will join him a week later. Staying for 5 days, not a lot of time! What are some must see places in a short amount of time that locals love and avoid the touristy stuff? We haven't decided where to stay exactly, we will probably start and end in London, but the other days our itinerary is wide open. Maybe make it up to Scotland if we can.

We enjoy shopping, old pubs, museums, sporting events, music, tours, etc. Due to my husband's health condition, we can't do anything like long hikes.

Any go to resources would be amazing!

Thanks in advance.