A new manager wouldn't necessarily improve the team instantly and as said, from Feb could be as little as three more games which they could have won, being liverpool, City and everton, of which considering the confidence and quality of the team, with no chance to buy anyone is a ridiculously tall order. Even the away Stoke and Arsenal games wouldn't have been walk overs with a new manager. The damage was done and I don't think it would make any difference.
Financially you're completely discounting the stock market. Fergie only announced his retirement effectively when Moyes was assured as a replacement. This was directly because of the effect on the stock market. If they fired him in Feb and didn't announce a replacement till late April or maybe May then the unknown situation could easily rock the stock boat and cause a hit WAY more than 18million, and way more than the Europa league could bring in, way way more.
They've according to that been talking to a few managers. So it to me suggests that like when Fergie announced his retirement and they said they would announce his successor basically by the weekend, to give investors a sense of control and planning going on. Moyes replacement likely needed to be fairly well assured before they announce Moyes being fired for the same reasons.
If you fire Mourinho and wait 2 months to appoint something new, nothing happens to Chelsea. if Utd fire Moyes and wait 3 months to name the manager to come in over the summer, the stock could plummet by 100million and never come back up. There is more at stake for Utd than other clubs in a similar situation. Likewise potential managerial replacements are more likely to be okay talking about leaving at this time of the season than later. Take Klopp as an example, in Feb he was in the champs league, in the league(barely but still in it) and announcing his leaving would be bad. He's out of both, but with a single final to go, announcing he's leaving could boost the players for the final, win one last trophy together kind of thing. Klopp/Dortmund could have refused to agree to sign a precontract 2 months ago but happy to do so today.