Here, based on nothing more than idle speculation, is the future of the Trump administration:
1) Mueller's investigation will continue. Trump would have to order the Justice Department to fire him. Sessions is recused. Rosenstein appointed Mueller and is not going to fire him. Trump would have to find someone in the Justice Department willing to do it leading to mass resignations, Saturday Night Massacre style, and that's hard to imagine. Plus, Congress would just reinstall Mueller to investigate under Congressional authority. So no firing. Or if there is a firing, no substantive impact on the investigation.
2) Mueller will continue to investigate Trump's financial dealings, which are the predicate for finding any quid pro quo payments (aka evidence of collusion). Trump's businesses and associates are questionable financial enterprises that use tactics (e.g. money laundering) that will not hold up to legal scrutiny. Purely speculation, granted, but like Frederick Douglas, it's getting noticed more and more. Trump's business associates may not have been involved in any collusion, but they are shady at best, and Mueller will have leverage on them to flip them against Trump.
3) Ultimately, Mueller will find evidence against Trump, if not of collusion in the Russia investigation, then of massive financial impropriety – if not current and ongoing criminal corruption – and will refer those findings to Congress; there is also the possibility of an indictment against Trump for said improprieties if they occurred before the time he was President.
4) There will be tremendous pressure on Congress to impeach prior to the midterms as Republicans' only defense against a massive Democratic wave election that would result in a Schumer Senate convicting an impeached Trump (and, if it gets that far, Pence), and installing newly elected Speaker Pelosi, 3rd in line for the Presidency, as President. (I expect we'll begin seeing evidence for Republicans jumping shipafter the Georgia special election next week if Ossoff wins (as I expect), especially if he wins by a larger than expected margin and Republican House members see a premonition of their futures.)
5) The safer (and more likely, in my estimation) alternative for Republicans, is for them to impeach/convict Trump and install Pence prior to the midterms rather than ride the Trump ghost-ship to their graves; more likely, they will convince Trump that they have the votes to impeach/convict and get him to resign and be pardoned by President Pence, and at least preserve the possibility of keeping their majorities. If so, Pence will have sufficient support post-midterms to stave off any investigation into his personal collusion with Russia. (Unless of course, he did collude with Russia, in which case all bets are off.)
There you have it – the future. Mock my words.