I'm not sure if the other persons insurance being valid or not will have any bearing on your liability.
Maybe you should give your insurance company a call to find out what is happening?
This thread may prove informative for you.
Your insurance company will now have no further involvement.
When the third party claim was submitted, your insurer would have made the third party's insurer aware of their drink drive covernance and your conviction. The third party's insurer would then have accepted that your insurer wasn't liable to pay.
So it will be the third party's insurance company who will pursue you via their solicitor. In fact, the third party can pursue you directly via a solicitor or represent themselves if they so wish. But it will almost certainly be the insurer via their legal team.
There is no time limit restriction so yes, they can begin legal proceedings at anytime, and they would be via the County Court.
Your insurance company are effectively out of this now, so contacting them will be of no use.
The third party claim could come at anytime. They won't have forgotten.
You should contact a solicitor regarding this, and make them aware of all the facts and communications.
The reality is that you are going to come home from work one day, and find a brown A4 envelope on the floor with your local County Court logo on the front. That will be the legal claim via a solicitor on behalf of the third party and their insurance company. At that point your are going to have to see a solicitor.
These things have a habit of hitting when you least expect it, and when you can least afford it. There is no value in speeding up the process, but you need to know where you stand legally, especially if you are looking to purchase a house, or if you have a substantial deposit sat in your bank account.
M