This is something I've pondered ever since all the headlines in the mid-2000s. Basically there was a lot of speculation that Tony Blair would try to go back on the Granita Pact (assuming, of course, it was the formalised deal that we all tend to think it was, but which the people involved tend to deny). That, rather than standing aside to let Gordon Brown follow him as Prime Minister, Blair would set up and endorse his own preferred successor. The chief candidate subject to this speculation was Alan Milburn, who was primed for this by becoming Secretary of State for Health in 1999--at a time when Blair was making the NHS one of the government's chief focuses for reform and increasing spending. However, much to everyone's surprise, Milburn resigned in 2003 "to spend more time with his family" and actually meant it. Although he was briefly brought back in 2004-5 as Chancellor of the Duchy of Lancaster in order to manage Labour's re-election campaign, he was never again considered a potential successor to Blair.
After Milburn, the speculation seemed to get increasingly desperate and seize on any figure who wasn't Gordon Brown as Blair's potential successor, often over-analysing any reshuffle Blair made as going towards pointing out this successor. I remember even John Reid dropping hints that he might see himself in such a role, which invited ridicule from the media. In any case, As We All Know, nothing ever came of it, and Brown was coronated as Labour leader on Blair's exit in 2007 without any challenger gaining enough MPs' support to force a formal election.
The upshot of this is, do we think that a non-Brown succession was ever a workable option? Say Milburn had managed to balance his family life (and, of course, actually wanted to be PM, which isn't a given). Say Blair planned his resignation in an organised way to prepare the way for him (that is probably more unlikely, as Blair like Thatcher tended to 'go on and on and on' by default). Say that Blair openly endorsed Milburn. What would happen? Presumably an open contest between Brown and Milburn. Who would win?
Assuming that this was ever more than just part of a propaganda war (as with Brown's "friends" dropping "hints" throughout the Iraq war period) and Milburn could have become leader and PM, what would his premiership be like, especially compared to Brown's OTL one? On the one hand, Milburn would certainly be a more capable PM than Brown (there are people in care homes who would be more capable PMs than Brown), but on the other hand, the economic crisis by this point was pretty inevitable and presumably shutting Brown out would create a lot of bad blood and divisions in the Labour Party that didn't exist so much in OTL, and which a competent opposition could potentially exploit.
Thoughts?