I don't doubt he would, which would mean that the Restoration would (if) occur under some VERY different circumstances? Perhaps at the head of an invading army? Parliament crushed beneath their boots? And how long before the next Civil War?
My thinking is that Charles I would
never give up trying to get 'his rightful throne' back, but again he was an idiot with a talent for making enemies out people that could have been his friends. So all in all he's wind up like 'the Old Pretender' and 'the Young Pretender' in RL. However he might be able to get continental backing for his ventures, after all what right-minded European monarch would even LIKE an English Republic?
All in all I could see Charles I's attempts to get 'Britain' back failing, though he might succeeded in landing in Britain a few times, maybe even seize London once or twice, but he'd never be able to really secure his holdings before being driven back by Cromwell's forces.
And due to Charles I being at the head of foreign forces, attacking his (so-called) nation, over and over again, I could see it doing A LOT of damage to the concept of monarchy in Britain. It would be even worse if/when one of his sons publicly became Catholic. Oliver Cromwell would REALLY milk that and use it to portray the Commonwealth as the ONLY bulwark against the return of the 'Tyrannical CATHOLIC Stuarts'.