Good question!

I think the politics would have been quite complicated – on the one hand, Tywin has been Hand for 15 years, he represents stability and competency in government, he’s the Crown’s major creditor, he’s got the army in hand de facto. It would be rather difficult for Rhaegar to refuse him in the moment.

On the other hand, Tywin’s power is in Rhaegar’s powers to dispose of, simply by naming someone else as Hand. (Indeed, if Rhaegar is really Machiavellian, he could publicly blame Tywin for causing Aerys’ death and really go to town on him.) 

On a third hand, it’s not clear how many alternatives to Cersei there were at the time…there’s no Arryns, 

In terms of the political effects…there are going to be quite a few Aerys loyalists who will run a rumor campaign that blame Tywin and Rhaegar for killing Aerys. I don’t think it’ll be given much credence by most (Rhaegar was quite popular among the smallfolk)…until something goes wrong and then people will start to whisper. 

There’s definitely going to be tension between Tywin’s faction and Rhaegar’s faction (Connington, Lonmouth, Mooton, the Daynes and the Whents) over access to offices. Rhaegar’s going to want his own men around him, and Tywin’s going to want jobs for Tygett and Gerion and Kevan and Illyn Payne and the like. And I would argue this tension would expand as other Houses get tired of Tywin’s monopoly on royal power – the Tyrells, the Martells, the Southron Ambitions bloc, these folks want their fair share of honors and offices and if they feel like A. Tywin’s never going to leave and B. he won’t share, you’re going to get some “evil councilors” politics brewing fast.