I've studied here in the past. I highly recommend their courses. They give their Saturdays over to Richmond Jazz School during the academic year. There are woodwind classes (as well as other instrumental classes) in the morning, split into two groups according to standard. Then there are a set of instrumental classes in the afternoon, where you can play with brass and a rhythm section. You can attend either or both of these classes over one, two or three terms.
The summer term class will be well through the course syllabus, so you might be expected to know a fair bit by this point. The tutors are helpful and will advise. More details at http://www.racc.ac.uk/richmondjazzschool.
The Easter jazz week should be a good introduction to this.
Easy access from west of London as well. Get on the M3 and keep going east.
Edit: the full courses also provide playing opportunities at The Bull's Head, Barnes.