Why are firms reluctant to interview you? Because they don't want to bear the cost of flying you over to London? If so, then you could take the initiative and get yourself over to London for a month or two. Of course, there's a risk that it will be a costly trip with no returns, but you have to decide whether that's a risk worth taking.And in any application that you make stress that you have the right to work in the UK - some recruiters will just assume that you don't and chuck your application automatically.
