First: good luck with that. The key states at the time (NY, PA, OH, NJ, MA) were comfortably in the hands of the GOP, as I recall.
Now, let's assume that Bryan somehow manages to pull it off. As inept as the Cleveland administration was at foreign affairs (recall that the ham-handedness of Cleveland's SecState Richard Olney almost bought the US a shooting war with the UK over a border dispute in South America), Bryan's would be worse yet. Bryan proved to be hopelessly naive when serving as Wilson's SecState; he was even less well schooled in that area in the late 19th century.
Likely the Maine would not have gone to Havana; pacifist Bryan wouldn't have allowed it. Thus, the Spanish grip on Cuba would have continued until conditions became intolerable for the locals, resulting in a nasty, mostly-guerilla-fought revolution of sorts. Cuba would probably become a nation run by a succession of dictators of one stripe or another, with conditions too volatile to encourage much investment. Puerto Rico might have followed a similar path. On the other hand, the Phillippines might have become the crown jewel of Germany's colonies.
I doubt Hawaii would have remained a sovereign nation for too much longer given the overwhelming US influence/presence: maybe until Bryan left office, but that's it.
Probably Bryan would have been a one-term president: his naivete and self-righteousness (he wasn't called the fundamentalist pope for nothing) would have made him annoying and a laughingstock to most voters. I suspect McKinley would have been elected in 1900, beginning a significant run of GOP presidents as a backlash to Bryan's ineptitude.