Hello, I have bought several properties in an around London and I wouldn't say that you need to use a London solicitor, I have used the same firm since I move upto the Capitol and I cannot fault them, they may not be the cheapest out there but cheaper than London firms by far.
Not sure if I'm meant to say names etc but I used a guy called (moderated) solicitors in Havant and I never had any issues with my purchases and sales both in and out of London.