The car was not insured for you to drive, so the insurance company won't help you.
As you are not yet resident in the UK you may find it difficult to find a claims solicitor who will work on a 'no win, no fee' basis.
You can get the details of the registered keeper of the other car from the DVLA - it will cost �3.
If the police are not charging the other driver with any motoring offences it is because there is not enough evidence. You will find it very difficult to win a private prosecution.
If the police do charge him, you will be in a better position to claim, although I suspect any pay out will be reduced because of the fact you shouldn't have been driving in the first place.
See a solicitor.