I've had two CDLs in the past, both with Co-op, they were very efficient at issuing the loans. I used them for training in the USA and converting back in the UK. I think most UK schools who are used to dealing with people doing commercial level flying will probably be registered. I believe you do now need to specify which schools you are doing which part of your training with.
However, bear in mind the bit about "extra expenses". If you list loads of extras like books, headset, travelling costs, accommodation, etc.etc. you can have a large part of the loan paid direct into your bank account.
Also, just for interest, the airline pilot on the front of the CDL pack is a real airline pilot and did take out a CDL for the training, there is a case study with the pack which explains it.