I'm planning to stay near Uppingham School and would really appreciate a personal recommendation of where to stay.
The Falcon is your best bet if you can get a room. The rooms are nice and reasonably priced.
The Garden Hotel is fairly nearby and the rooms are reasonably priced. It depends on if you are there for business or a short break. I can't see whether they have wifi access in the hotel which might be a pain if you are planning on working or answering emails while you're there.