This topic of cases is well worth a search.
I am from the UK and I go back at least a couple of times a year. Big problem with rigid bike cases is ground transport. Black cabs can't take them so it can be a mission. Depending on your frame material I would think hard about getting something like this:
Although people use this for their carbon bikes I don't
London isn't a bad city for cycling but cycling down to the south coast (somewhere like Brighton) or getting a train north to places like Cambridge is always fun.
Ride safe and ride on the left!