You'll have 90 days to use your visa from the day it is issued. So that means anytime within the 90 days before you travel. Generally about 1 month ahead is fine. And London is known as one of the fastest Thai embassies in the world, normally processing applications in 2-3 days.
Either consulate in Laos is probably the fastest option. 2-4 business days after you pay at the embassy. If you want it completely online, then probably Vietnam.