If Cromwell isn't involved in the English Civil War, I have a feeling it would turn out the same way (Charles imprisoned, Parliament and the Army victorious). I don't think Fairfax would be able to impose a settlement like that on the country though- the New Model Army was fanatically Puritan, with preachers assigned to every unit, and many commanders preached their own sermons to the troops- so I can't see them accepting Presbyterianism as the new Church.
OTL Fairfax seemed unwilling to stand in the way of the Army's politics, staying out of discussions about the monarchy's future, and physically leaving the Army when it became clear Charles I was to be executed. I don't know who would replace him, but without a very political general firmly in charge, its possible that political radicals (Levellers mainly) are able to expand their ranks in the Army.
Another possibility is that without a political general like Cromwell new elections are actually held sometime between '48 and '50. There seemed to be momentum toward that, with Pride's Purge and the events afterward derailing it. The resulting Parliament would have more legitimacy, especially if some kind of re-districting had been done prior to the elections, and that could combine with a non-political Fairfax to keep the Army out of politics.