TR wasn't as Anglophilic as Wilson, who I think was one of, maybe the, biggest Anglophiles ever in the Oval Office (he preferred the parliamentary system to the Constitutional one), but TR was more of a warmonger.
For TR as warmonger, see my previous message.
Re Wilson, he may have been an anglophile, but before all else he was a Wilsonophile. Afaics, for all the German provocations, his real reason for going to war was fear of being left out of the peace conference - because he had convinced himself that only he could get the peace right. Two years later, despite being clearly incapable of performing his duties, he would cling to office from a similar conviction that only he could get the League of Nations question right. In fact, it's not obvious that either the ToV or the 1919/20 LoN debate really benefited much from his involvement, but he'd got this vision of himself as the Indispensible Man.
President Bryan or La Fallette is what you want.
I don't really see how LaFollette could get in. The GOP would never nominate him. The only possible route would be the one that put TR in - becoming VP followed by his predecessor's death in office - but even that seems highly unlikely.
President Bryan is more of a possibility. What you need is something - illness, maybe? - to keep him out of the 1908 race. It won't make much immediate difference - whoever replaces him on the ticket will lose to Taft as he did - but without this third defeat, he is front-runner for the 1912 nomination, and with it the Presidency. Of course, he would need to be re-elected in 1916[1], but if his domestic policy (the decisive factor that year) is the same as Wilson's there is no obvious reason why he shouldn't be.
[1] Assuming there is a 1916 election at all. If the single term amendment goes through (Bryan won't block it as Wilson did) then the next election may not be until 1918.