No they won't as they've started the claim they have the legal right to take over and complete (subrogtion I think, might be wrong, the word escapes me). As they've started no 3rd party solicitor will be interested in getting involved unless paid directly.
For future reference for anyone, if a none fault claim always persue the 3rd party insurer direct, it can sometime be more hassle, but if you know what you're doing you've far more clout. If you go through your own then you'll be bound by their T&C's