I don't think that has anything to do with medical treatment. Are you trying to solve some other issue?
For what it's worth, I think the way consequences work in play is fine. But some deeper healing rules would be nice.
Changing the Consequences can result in different healing mechanics entirely. So instead of treating consequences as almost purely stress affecting things, maybe players and GMs should be treating each consequence as a sort of mini-story, starting with how they acquired and ending with how they get dropped. The best part of this is that could fit in fine with the original rules for Consequences getting healed, as the table could decide whether to actively pursue healing for a character, or to just wait it out.
So instead of PC Bob taking a mild and moderate during a chase scene and just waiting for them to go away, or even just heading to some specific location to get treatment (followed by more waiting or more stress), it could work a little more like this.
If a consequence on one particular character is a problem for the party (main combatant can't fight, main talker can't be seen in public, main caster is talking gibberish all the time instead of casting), the party can choose to single out that consequence and work it off more quickly.
So instead of waiting like normal, Bob buys some in game time to heal up with skills/tags/Party help/A fate point, and uses [whatever time period the GM declares realistic] in game to work with somebody with the Doctor stunt to try and un-mess up his arm.
While any number of mechanics could be used during that time to represent healing, what could work best here is to establish some timeframe for the number of rolls allowed and a number of shifts necessary to overcome the consequence. But some of the other suggestions made so far could also work, such as the each consequence providing a point of armor until it's really gone or providing extra boxes to take stress, as could a GM just deciding that nothing needs to be done here; it's up to the other characters to buy in game time to make up for the real life wait for a Consequence to go away.
The point isn't the specifics of how a Consequence is dealt with, it's how the idea of healing is treated as gameplay element. I think in FATE systems, dropping a Consequence requires not some special point by point 'how we get rid of this thing' treatment, but rather an acknowledgement and concerted effort by the players through established rules or possibly simple rules variations to overcome that particular part of the plot. Or, in books, movies and TV shows injuries work as the plot and themes demand, and FATE should emulate that rather than simulation of real injury.
During that time the Bad Guys are still doing things, so there's an ongoing contest (like, say the Cat and Mouse contest from YS pg 195)for the other PCs to deal with in keeping the Bad Guys resources tied up while Bob is trying to get back to functionality. That way, although Bob is gonna get back to main plot with a clear track more quickly than normal, there's a trade off for getting Bob healed as alternative to playing on with an injured Bob. Namely, the party becomes a sort of shield against further harm for Bob and the things that would compel Bob to get back out there so that Bob and his doctor can get rid of that really annoying aspect. Individual PCs could also skip out on helping out with the contest and go help Bob instead, if that's necessary.
Also, Tarans' idea for tags that could be used to pay off Consequence compels could fit in perfectly as an lower effort narrative mechanism for medical treatment. This way, should a Consequence become a problem for the players, they can slap a band aid on it, or focus on treating it like
any other in game problem and just work together on fixing it.