During the offline period for Westminsters laser cutter i did some cutting at Central St. Martins Art College in Holborn.
It's £30 for the first hour, £15 thereafter (with a student card) and the technician, Ben Knezevich, will look after your project from start to finish. The charged timing is also 'flexible'- probably as long as you are decent, patient and friendly!
You can book a time slot on 020 7514 2147. The effective cut area is large at 1400 x 900mm. However, the machine has a max material cut somewhere around 2/3mm ply. In which case it takes a lot longer and costs a lot more!