Do you have any poultry vets, like a livestock vet who treats the poultry birds? This is better than a traditional or general vet, as a chicken is very close in anatomy/physiology to any parrot...Although, I also don't understand why any vet at all can't do a culture and sensitivity on both your birds???? All they do is take a swab of their nostrils/throat and to a normal culture/sensitivity, the same as they would on any dog or cat who has an infection...This would diagnose the exact microbe that is causing the infection, either the specific bacteria or fungi, and would also tell them exactly what antibiotic and/or anti-fungal medicine the microbe would be sensitive to...They do this on dogs and cats every day, and it's no different...They should also do a Fecal culture on their droppings as well..
As far as your relationship with your birds...This is exactly why you should never, ever bring home a second bird for your first bird, because you think it's "lonely" or it "needs a friend"...Your first Budgie did not lose trust in you at all, nor did his relationship deteriorating have anything at all to do with the fact that he wasn't fully-tamed when you brought the new bird home...
Regardless of what type of bird you have, how long you've had them, or how close your bond is with them (you could have a Macaw that you've had since it was hatched, hand-fed it yourself, and have the closest bond with the bird possible for years and years, and this could happen), whenever you bring-home another bird, specifically a bird of the same species and of the opposite sex, you run the risk of completely losing your bond/relationship with your bird...They may absolutely hate each other and attach/hurt each other, they may simply tolerate each other but not really like each other, they may like each other a lot but not bond closely enough to mate, or they may bond extremely closely and if of the opposite sex may start mating...These are the possibilities, and there's absolutely no way to know what will happen until you put them together..So it seems that when you brought home the second Budgies, they bonded very closely with each other...And when two birds bond very closely with each other, they tend to dissolve any close bonds/relationships with human beings. It has nothing at all to do with "earning his trust back" at all, and nothing to do with the fact that he wasn't fully hand-tamed when you brought the second bird home, as I said, you could have been spending every moment of every day with him attached to you and cuddly with you, and still when you brought that second Budgie home he would have totally bailed on you and wanted nothing to do with you...
Why does this happen? Well, it's pretty simple actually, because they much prefer a close bond/relationship with one of their own species over a human...You said that you thought your first Budgie was "lonely" and "needed a friend", which was why you brought the second Budgie home, but you were supposed to be his "friend" and keep him from being lonely. If you form a close bond with a bird as "it's person", and you spend a good amount of time with them every day, then they don't need another bird to be their friend...And you see what can possibly happen and usually does happen if you do bring home another bird for them...
So the answer to your question is quite simple, as Owlet already explained..If you want to have a close bond or relationship with either of your Budgies at this point, the only way to have that happen is to completely separate them into their own cages, with lots of their own toys, their own food/water, etc. And you'll also most-likely have to keep them in separate rooms too, as if you have them in separate cages but in the same room so that they can see each other, they will most-likely just cry all day long to be with each other...The other issue you're going to run-into here if they are an opposite-sex pair with such a close bond and living in the same cage is that they will most-likely start mating eventually, and the last thing you want are baby birds in an environment that is making them sick...So you essentially need to break the bond between the two birds...
However, if you're going to do this, then you really need to make a commitment to spending a large amount of time each and every day with both birds separately, as first of all you'll not be able to bond closely with either of them if you don't do this, and it's going to take time for a bond to develop with each of them, so you must have patience as well, because it could take months and months...Plus, it's not fair to either of them at this point if you separate them and then don't commit to spending a large amount of time with each of them on a daily basis. But if you're willing to make that large a commitment to each of them, then that's what you need to do..
Something else that you might want to consider, and obviously this is totally up to you, but you may want to think about re-homing the second bird to another responsible, loving home that will spend ample daily time with her. This will ensure that both birds will get lots and lots of time and affection every day, and that each can form strong bonds with their "people" or "person"...And if you want to have that type of relationship with your male bird that you've had the longest, then this might be the best thing you can do, for both the birds and for yourself...Just a suggestion.