Well if Oliver had lived longer, or if his son had been more up to the job it might have lasted a bit longer. The problem was that the Stuarts had to much left over support, and once Cromwell was dead and Richard was not up to the job, your back with the Rump/long Parliment. The Long was not very popular by this point, and was elected nearly twenty years before. Meanwhile with Cromwell dead all the resentments he'd suppressed while alive (usually by main force) burst back out, the cavilliars gained the support of the army and Charles comes back.
If Oliver lives longer, and plans for what happens after his own death, and maybe...maybe if Charles has a (suspious yet utterly plausable) accident, then the commonwealth has a chance, especially if the only alternate is James who is already flirting with Catholism and is even more fond of absolutism than Charles II.
Charles II was moderate enough that he could get back the throne when the country was desperate and Olivers death had created a crisis he could exploit, James might be able to do the same thing, but given his views on religion you could easily risk the Glorious revolution twenty years early.
The commonwealth was unstable from the start Scotland and Ireland had only joined at gunpoint, and the English Republic itself was put together in a matter of days in 1649. That combined with years of war, and half a dozen failed experiments in the form of government left people fed up and nostagic for the rule of the Kings (which had at least been fairly stable). That let Charles come back, have a quarter of a century of (mostly) stable rule, and then James reignited the issues and forced the working out of a solution most sides could live with.
I don't know if King Oliver would have made a difference, but if he'd said "In the name of God go!" a year or two earlier, and if he'd lived another five years then the stable later Cromwellian commonwealth might have had time to bed in, before the issue of succession arose.
A dead king or a live protector, either would have at least bought time.