I've often thought that there's an outside chance (veeeery outside) that Blair is still PM without Iraq.
FWIW, I think a lot of the intra-party hostility would've played out like it did OTL. I think there's a chance he might have wanted to stay on until 2008/09 to beat Thatcher's record. Of course, that presents its own problems because the economic crisis would still have occurred (assuming no other PODs), which is obviously a problem for Labour in any general election. The response to that crisis might have been very different without Brown's leadership at the crucial time - I think Blair has spoken since about how he would've pursued deflationary measures at the time but that might just be because he wanted a way to criticise Brown (who knows with those guys) - so that would've been very different (worse IMO but that's something people can disagree about). In that context, we might end up with something like a Cameron-lead pro-austerity government in 2010 anyway.
Of course, if your POD is a Gore win in 2000 then obviously the international scene would be very different so Blair might be very different in response to that. (Blair's love-in with Bush was basically conditioned, as all things in the Labour party are, by his response to the defeat in 1983: namely that he had to attach himself to whoever the American president was. A different president in '00 and '04 would probably make his foreign policy very different.) My one speculation is that, without the discrediting of liberal interventionalism due to Iraq (although that would depend on how Afghanistan goes), Britain might have more seriously contemplated intervention in Zimbabwe in c.2008.