I agree that if you want to see Hague in office, you need him not to be the Conservative leader from 1997-2001 and instead, he should become leader after the 2005 election.
If this happens, I see him being to Cameron's right on most issues, though probably still open to a deal with the Lib Dems in the event of a hung parliament (he was associated with the SDP at one stage, apparently).
He'd be moderately Euroskeptic-though as he's Foreign Sec currently IOTL, I don't see much changing in that regard. One advantage he'll have in comparison to Cameron will be the lack of a "Toff" image.
I'd imagine Cameron being fairly high up in the Cabenet, though I don't see Osborn being as prominent somehow.
If you want an earlier Hague Premiership (Eg in 2001), your best bet would be to have Kinnock win in 1992. The Tories stand a good chance of winning in 1997, lead preferably by someone who won't want to stay on long-term (perhaps Hurd? Haseltine? or perhaps Major stays on after the 1992 defeat if it's narrow enough??). Hague works his way up the Shadow cabinet (and the cabinet) in the years 1992-2001 (just about doable imho) and when whoever's in power steps down in 2001, Hague could be in a good position to take the reigns (I see Portillo as a more obvious Thatcherright successor, but I suppose him self-distructing in some way could be possible).
Or you could have Kinnock win in 1997 despite Black Wednesday (unlike a lot of people I don't see 1997 as an obvious Tory win in a Kinnock wins TL, though imho it'd be pretty likely). So Kinnock and co are back in for another term by the skin of their teeth and the Tory leader gets the blaim for it, leading to Hague's assension to the leadership.
The fuel crisis of 2000 goes worse for a dying Kinnock government than it did IOTL (butterflies could mean it happens just before a theoretical General Election as well) and Hague wins despite the good economy with a comfortable majority.
In these circumstances, I still see the boom happening, though it might be less severe than OTL (and therefore we could be spared the worst excesses of the crash). In Foreign issues, I see Hague having a close working relationship with Bush (assuming butterflies don't make Gore-or someone else president) but he'd be more distant than Blair was OTL.