@eb424 you could try Richmond Jazz school, they do a series of courses on Saturday from beginner to advanced. The makeup of the band depends upon who signs up so you may or may not have a full rhythm section although you will certainly have saxophones. Try the beginners group here
Jazz Ensemble 1. Make sure you don’t get sent to the Jazz Performance Ensemble, different beastie, reading definitely needed, and not what you want.
If you're interested I would call and see if you can drop in as an observer and get a sense of whether it's what you want or not. I think this term runs for another 5-6 weeks. I've attended a couple of these over the years, Julian Costello does a good course at different levels but that's on Wednesday so probably no good for you though you could probably find one to fit you if you can make it.
I haven't any experience of the beginners on Saturday but have attended the advanced and they work you hard and you learn things you didn't know you needed. Plus you get to play as an ensemble. The teachers are all working musicians. Just a thought as it's a lot closer to you than Chalk Farm and they have parking. I think they also do a Summer School if you're feeling brave.