My experience is most furniture making courses are run outside of London - cost of space is too much in the capital.
There’s a few schools around the country
I met MarcFish and nice guy, design and contemporary edge to their work.
Most do 1, 2 day or week long tasters. They are expensive though and you’d need to relocate.
More locally try neighbouring adult ed courses, they are very cost effective - the Southwark one I did had an excellent workshop facilities.
If you can wait till summer there is a Japanese master woodworker coming to give a month long course in London. He usually teaches the course for 4 students at a time in his Kyoto workshop. Its focused on japanese handtools and joints. With those skills you could make any piece of furniture.
I’ve been thinking of bringing in a furniture maker to run courses at SLMS for members, if there’s demand we could look into that more. Benefit is it would be tailor made, local and lower cost.