There aren't commercial breaks on the BBC, it's funded by an annual license fee, so programmes generally run without interruption.
And yes, tea is that popular. In my office, each person will typically drink 3-4 mugs of tea during the day. Many people continue with at least two mugs when they get home.
I don't know about Top Gear, but Doctor Who is cut on reruns. It is broadcast in its entirety, albeit interrupted by commercials, on first showings (which means it runs long by about 15 minutes).
It really depends. Quite a lot of older people drink only tea. Young professionals living in London drink espressos, capuccinos, that sort of thing. I don't know anyone who drinks instant coffee (although I'm sure some people do).
And yes, tea is that popular. In my office, each person will typically drink 3-4 mugs of tea during the day. Many people continue with at least two mugs when they get home.