Here's a brief summary of the UCI and its current organisation (taken from its constitution, with a few comments and interpretations), and some vague ideas for change.
The UCI claims sole global authority over the sport of cycling. In terms of organisation, the UCI is given its mandate by 170 National Federations (NFs) although (just like in football), there are also five Continental Confederations (CCs), which have a rather unclear role. It gets funding from these national associations and from event organisers and teams who have to pay registration fees (which vary depending on the level). However it only represents the National Federations, not event organizers, teams or riders.
Representatives of the National Federations form the UCI Congress. The Congress elects the President for five years at a time. Just like the FIFA elections, this is always a controversial process with allegations of favouritism, bribery, impossible promises etc. etc.
The President is in theory still responsible to the Congress, although in practice he pretty much does what he wants with the conivance of the Management Committee (MC), which is responsible for the day-to-day running of the UCI. There are fifteen full members of MC: ten are elected by Congress, the other five are the heads of the CCs. There are also a few other non-voting members. Finance is dealt with by a nominally independent person selected by the MC. There are also Commissions on particular issues.
Because of the lack of input of event organizers, teams and riders, a Professional Cycling Council (PCC), was set up in 2000. This has ten members: 6 chosen by the MC, 2 by the race-organizer's association, 2 by the teams' assocation and 2 by the riders' association. The PCC is responsible for race rules, but otherwise has nothing but advisory power. As you can see too, despite the sop to riders, teams and race-organizers, the PCC is still controlled by the MC delegates.
In my view, the first thing that needs to change is to increase better balance the role of the National Federations, Race-Organizers, Teams and Riders. My view would be that the PCC should be reformed to include no MC members and more from the other sections and given some kind of power-sharing role with the MC. Both should be able to make proposals to the Congress, for example. Teams, race-organizers and riders should also get votes (somehow) in electing the President. I haven't thought much about how this latter idea would work.