Okay, so if it turns out there are no appointments and you can only get one 45 business days ahead, which will be too close to when you go to the US, it might be worth going ahead and applying by post ASAP.
You have 10 weeks before you go to the US, which might just be enough time to get your application processed by post, as long as the processing times don't stretch longer than 8 weeks.
I would keep trying all the different premium centres though, just to see if any appointments pop up... you may find you cannot get an appointment in London though and may have to travel to whichever city you can get an appointment in.