It clearly depends on how Abramovich would get along with Sir Alex. Abramovich seems to be extremely jumpy when it comes to his favourite kind of football: Villas-Boas' style was glamourous, but not guaranteed to win a major trophy, so he sacked him for di Matteo. The latter won the Champions League, but apparently Abramobich wasn't happy with his style of football, so sacked him also. I think such an attitute would clash with Fergie.
The ultimate question is though, would Roman be stupid enough to sack Fergie considering his history with Manchester United, I would imagine it would make Ashley's sacking of Kevin Keegan seem like a popular decision
Back in 2004, Mourinho became head coach/manager of Chelsea, which led to a Euro shopping spree that included the likes of Cech, Robben, Carvalho, Ferreira, Mutu and Drogba. With a similar thing happening at Old Trafford at around the same time, the natural progression of the United squad wouldn't have happened in the same way. So maybe there would have been a short-term success in 2005/06, but with dangers of a bad team morale. Rooney and CR7 might be sidelined in those years and then sold on - the same goes for the likes of Fletcher, Carrick, Ferdinand and maybe even Giggs and Scholes, if they have more competition from foreign players.
In its first 3-4 seasons (2003-2004, 2004-2005, 2005-2006 and 2006-2007) under Romans ownership Chelsea spent £330.7 million on 34 players. Manchester United on the other hand during this time spent £100.3 million on 18 players.
During this time here is how it turned out in terms of results:
Chelsea
2003-2004: EPL: 2nd - CL: Semi-Finals - FAC: Fifth Round - LC: Quarter-Finals
2004-2005: EPL: Winners - CL: Semi-Finals - FAC: Fifth Round - LC: Winners
2005-2006: EPL: Winners - CL: Knockout Round - FAC: Semi-Finals - LC: Third Round
2006-2007: EPL: 2nd - CL: Semi-Finals - FAC: Winners - LC: Winners
Manchester United
2003-2004: EPL: 3rd - CL: Knockout Round - FAC: Winners - LC: Fourth Round
2004-2005: EPL: 3rd - CL: Knockout Round - FAC: Runners Up - LC: Semi-Finals
2005-2006: EPL: 2nd - CL: Group Stage - FAC: Fifth Round - LC: Winners
2006-2007: EPL: 1st - CL: Semi-Finals - FAC: Runners Up - LC: Fourth Round
Looking at those results I would say that Manchester United could win the EPL or at least come second in 2003-2004 and all but win it in 2004-2005 and 2005-2006. They would perhaps reach the final of the Champions League (or even win it) and win the FA Cup as well.
Beyond that though there lies the question, which players end up in the team ITTL. Some have suggested that Rooney and Ronaldo might not end up playing for the Red Devils at all as might Carrick. Likewise was there ever a time that there have been suggestions that Giggs and Scholes might go and lastly which defender at this time could leave Ferdinand on the benches?
Without an oligarch-backed Chelsea, the top duel would still be Arsenal v United. But I guess around 2008 or 2009, Liverpool might finally have a go at an EPL title as well.
Especially if the Hicks-Gillette takeover is butterflied...