Well even if Richard had become King of England, it wouldn't have mattered much as he wasn't up to the job.
Now whether Oliver, in his last days, permits a large number of citizens to vote for parliament is possible, but it'd be in contrast to what he did during & after the Putney Debates.
I'd say the best bet for a Cromwellian Royal family was if, when Richard abdicates, Henry Cromwell quickly returns from Ireland & establishes himself as the new Lord Protector in London. I'd say the Royalists will still have a go at invasion, which Henry will defeat (as he's a chip off the old block unlike Richard), thus firmly establishing himself. Then, a year or two later, Henry takes the Throne as Henry IX.