Ok, I have done what you want to do.
You do not need to submit your log book to the CAA to see what you need to do. Give them the least opportunity to lose stuff. Don't worry, they'll get a chance to look at it later.
Medical. You won't need to do the full initial class 1. The Medical branch is quite helpful and friendly.
Before shifting to the UK, I would get 500 hours on the 737 - especially if someone else is paying for you to do so. You can add sim time, but, to keep it easy, I'd just get the 500 hours flying time. Is it the NG? Easier if it is.
Also, I would get a letter from your employer on letter head paper stating your dates of employment and logged time with them. (I did this, CAA accepted it)
While you're enjoying all that fun, do the Bristol Ground School theory for those wonderful fun 14 EASA/JAA exams. You'll love them. They are just so much fun.
Now all you need to do is an observed LST (licence skills test) with a TRE and a CAA inspector. As you have 500 hours on the B737, this is the ideal type. Believe it or not, this is the hard part.
You have to arrange all of this yourself. I would definitely get practice in for the LST, the UK have slightly different speeds for the Non precision approach than Australia.
If you want to use a different aircraft type, you'll need to do the full type rating followed by the LST with the CAA observing. (Ironically, this is definitely easier to arrange)
Once you have this done, you can then open a EASA/JAA ATPL.
I haven't seen the abbreviation CS25 but it looks equivalent to JAR 25/FAR 25 or as we know (knew?) it CAO 20.7.1b.