Would Labour have performed as well in future elections?
They probably wouldn't get the huge majority that Blair did in 1997. But by 1994, the Conservatives were always going to lose that election comfortably to a semi competent challenger. They would still likely get something in the upper 300 range. Still the largest Labour majority since 1945. Portillo would probably survive in this scenario though, and take over as Tory leader in 1997. Brown probably wins another majority, but maybe something more like 2005 than 2001.
How long would Brown stay in office?
Hard to say, but I think Brown would be PM for a shorter period than Blair, if anything because the latter represents a more attractive alternative electorally to the former, and with a reduced majority, there might be more of a temptation to stick to the original plan, with Brown going around 2004.
What role would Tony Blair play? Would he stay in home, move to chancellor or run for deputy leader?
He wanted to run for Deputy on a joint ticket with Brown in 1992, so it's entirely possible he actually does it in 1994. Blair could have taken on any of the three other great offices of state. He had taken on shadow economic portfolios in opposition, so he could be Chancellor if he asked too be. If it wasn't him, it's hard to think who it would be. Brown was on bad terms with Robin Cook, so it wouldn't be him. Maybe Jack Cunningham? He was getting on a bit, but he was a right winger who was close to John Smith and had shadowed trade and industry for a while.
Would we see an similar pivot towards New Labour?
Yes, Brown believed in New Labour as much as Blair. He actually had stronger ties to Peter Mandelson than Blair did to start off with.
What would Browns relations with the US and EU be like? Would he still go to war in Iraq?
I can't see Brown being as close to GWB. Blair seemed to be the main instigator of the whole Iraq thing, and Brown was fairly apathetic. I don't think he would have risked so much of his personal reputation on invading Iraq, like Blair did. He'd probably go into Afghanistan though. Britain's stance toward the Euro would be broadly similar if Blair had a similar power sharing arrangement with Brown, except Brown would be PM and Blair would be the senior cabinet member in this scenario. A fudge would be agreed, the UK wouldn't join in Brown's premiership, and by the time he's gone it would probably be too late for Blair to successfully make the case as PM.
When Brown stood down would Blair be able to replace him or be outflanked by one of Browns prodigies?
No, because Blair was skillful and charismatic enough to not let that happen, and there weren't ever really many Brownites who were particularly impressive public performers. I could see the accession of Blair being met with a bit more apathy from the public, though, since he would have been an established figure by the time that it happened and they would have quite a while to prepare themselves for it.
But most importantly what impact would Brown going first have on British politics?
I think Blair takes power in 2004, probably beats Hague or someone with a similar level majority to 2005, but then goes onto lose the next election because of the financial crisis. Maybe he could salvage enough seats to forge a workable coalition with the Lib Dems. I don't think Blair would have been quite so discredited in this TL without Iraq. In fact, due to the economically interventionist measures he might be forced to take due to the pressures of the financial crisis, he might be viewed as a more left wing PM than Brown, and though he'd attract criticism from the left, he'd still be respected within Labour, much like Brown is now. Labour would probably stay in the centre for longer under David Miliband or someone like that.