We recently had a delivery driver injure himself on his own lorry whilst on site.  He accepted first aid treatment but refused to go to hospital to be checked over.  If he had later had an accident on the road in the vehicle, would we be liable in any way?  We did try to get him to go to hospital as a precaution for a laceration but he said he was fine with a dressing.  He isn't a British citizen.

No - you wouldn't be liable it is up to the driver themselves to ensure that they are fit to drive, so long as you have a record of the treatment and the advice given it then becomes their responsibility.

Not your problem. He took the responsibility upon himself to operate his vehicle in the condition he was in.

If he had an accident as a result of his earlier injuries he would actually be in it for using a vehicle when he was impaired.