I can only speak to rentals, but six weeks out isn't too bad, but it's a little early. We found our current place about that far out, though, as did a few friends. The whole process is such a crap shoot and comes down to dumb luck much of the time.
You do get more bargaining room closer to a move-in date, but that can also mean inventory has been snapped up. Our current place had been on the market for a while and priced too high, so we were able to negotiate on several points.
I'd definitely try to see some places while you're here next week. We had a much lower price point than you for rent, so I have no doubt (or at least a strong hope) you'll be saved some of the awfulness I saw, but it'll still give you a good stating point on what's acceptable for you and what's not.
Good luck with your trip. Le Cordon Bleu is only a block off of Marylebone High Street, so spend some time walking around and getting a feel for it. It's a wonderful neighborhood and I really love it here.