Wiki says Cromwell "agonised for six weeks over the offer", perhaps he wasn't as staunchly anti-monarchy (at least as long as he was in charge) as he is sometimes portrayed.
True, though in the end he was a soldier, and I've read that he was very nearly convinced to side with Charles I during the civil war.
I think this might be one of those occasions where wiki is less than reliable.
Did Cromwell consider it? Yes. Of course he did. Monarchy was the de-facto model of rulership at the time. Various groups did beg him to take the throne for the sake of stability.
Did he agonise over it for weeks? Its hard to say. We've no idea when he started considering it or not so timeframes are iffy.
A lot of the accounts we have of this period are from the Victorian period where historians were at pains to stress how much of a good guy Cromwell could be. Some of those accounts of his decision-making smack a little of that standard trope of 'It took you ages and ages but in the end you made the right call' type of thing. A dramatic flair to the process of being an upstanding historical character.
We've no real way of knowing his thought processes, so I think its a little unrealistic to say 'No this requires a completely different Cromwell'. It is unlikely but the man, like many at the time, was driven by a passionate belief in divine providence and so its not impossible he would have taken the throne if he felt God willed it.
To speculate on the OP's idea:
@ You might see a series of minor rebellions and upsets, particularly within the New Model Army. Several Colonels recorded in their memoirs that they confronted Cromwell in this period and threatened to rebel. Whether they would, mind you, is another thing.
@ In the event of any uprising Cromwell would likely win. By this period the nation was utterly sick of civil war. Just think how few people turned out to support Charles II at Worcester in 1651. Plus Cromwell was an able commander.
@ Beyond that I suspect it would likely go as OTL. There would be very little difference to the governing of England - regardless of the title he used Cromwell was firmly in power in 1657.