You should have no problem with making a successful claim against the third party, but I should point out that it is not something that is going to be settled overnight, you will be looking at several months at the very least.
You have three years from date of injury to issue proceedings against the third party, although it is very rare for a case to get to that stage and go to trial.
In the first instance, I would suggest you speak to your insurance company as you may have legal expenses insurance and they will instruct one of their panel law firms to represent you (not the best choice but will certainly get the ball rolling), if you do not have legal expenses, then speak to a local personal injury law firm as they will be able to represent you at no cost to yourself on a conditional fee agreement (no win no fee)