Whats a good beer to have with steak and mashed potatoes? What type of beer goes well with hearty, 'British' style dinners?  What characteristics should I look for?
This of course is pretty subjective, but I will try to answer it anyway.  

I would say that the choice of beer in this case should depend on one of this factors:


You want to accompany or match the taste of the steak and seasoning.
You want to have the beer as a regular drink nearby that does not disturb the taste of the 
steak.


I will assume that you only use salt and pepper for seasoning.  

Case 1:  

For this, I would recommend a Dunkel (dark lager). From my experience, it compliments the taste of the steak and seasoning, without disturbing it.
The taste does not sit in your mouth too long, thus if you wait a moment (few seconds) between beer and the mashed potatoes, it should not cause any trouble with them, too.

Case 2:

If the beer is just meant as a drink nearby that should not disturb the taste of the meat, any lighter Pils (pale lager) should be fine.
It does not have a long lasting taste, goes well with mashed potato and does not clash with the steak.

Disclaimer:
I am from Germany, thus my choices might be a bit... regional. All of the above are just personal recommendations, based on my experience.