You haven't mentioned budget or type of venue but you could try Roast in Borough market (not actually been - but heard good reviews - expensive though). Trinity pub on Borough Hi St next to Borough tube is very nice (have been several times) and good prices. I'm sure a lot of the pubs will take bookings. Charles Dickens pub in Union Street also very good - they'll take bookings.
I have been once and was impressed with the menu but sadly, ended up ordering the wrong thing and it wasn't good at all (Sorry can't remember what it was. I have obviously blocked it from my mind but it wasn't one of the roasts) but everyone else chose well and was very happy.
Ignore the child-baiting. It's becoming quite popular now that they're cracking down a lot more on racism and sexism. As we all know, the only people who might possibly get rowdy in pubs are children.
In East SE1, for £15 a head, you can probably still do the deal at the Chop House bar. Drinks aren't cheap, but imho it's a nice way to have good food and service in a nice setting and not pay the earth.
Or, in West SE1 The Goldsmith does an excellent Sunday lunch, and keeps getting good reviews on here.
Not sure what either place does in the way of childrens' menus, if that's important. Good luck. Do tell us where you went and whether it was any good, if you don't mind.