Hanging Cromwell and other Puritans definitely isn't going to get a long lasting Republic in England. Somehow, the Commonwealth set up after Charles I got beheaded is going to have to avoid a few things
- growing animosity between the Parliament and the army
- the growing animosity between moderates who wanted to reform the Church of England along the Presbyterian model, the "Independents" who wanted largely self-governing congregations, and the even more radical groups who endorsed a variety of extreme religious (and often social) views
These tensions led in OTL to Parliament first expelling the real moderates, and ultimately having the "Rump" be disbanded by Cromwell with the support of much of the army leadership. If you can somehow find a way to reduce these tensions, and marginalize the more radical groups, you might get a situation where the different political and religious factions hold together long enough to set up a viable system of government.
The only other way I can think of is a later POD - when Oliver Cromwell dies, somehow he gets a very politically astute successor (perhaps he doesn't go with his son but chooses one of his top subordinates) who calls together a new Parliament and sets up a system where an elected Parliament and the Lord Protector share power, while at the same time keeping the army happy as well.