The easiest and most economical way of getting to South East Asia from the UK is to fly, though you can reach the region by other means too, like by train or boat.
The vast majority of people reach South East Asia via a flight to either Bangkok or Singapore, both of which can be flown to direct from London. The capital of Malaysia, Kula Lumpur is also a popular entry point, as is Jakarta, the capital of Indonesia. Other major cities which receive international flights are Hanoi, capital of Vietnam, Phnom Penh, capital of Cambodia, and Phuket, the largest of the Thai islands.
Journey time to Bangkok is around 12 hours direct, though keep in mind some of the cheaper fares will include a stopover in somewhere like Dubai for a few hours, which can really pile on the time it takes. Still, though, many think that’s a sacrifice well worth making.
It’s possible to reach South East Asia by train, but the only route available is into Vietnam from China. You can actually travel all the way to this entry point by train from Europe, via the Trans-Siberian Express from Moscow, which is an epic travelling experience in its own right and certainly makes for a better story than hopping on a plane. However it is more expensive than flying in the long run and will obviously take considerably longer.
It’s possible to reach South East Asia via the sea, but this is reserved almost exclusively for those on cruise ships, so if this doesn’t feature in your gap year plans then you will need to consider one of the above two options.