I can't really say for sure on how long it would take for you to trust them. I, by nature am not a very trusting person but can usually figure out if I can trust people fairly quickly. If the therapy bill is being payed by the military I would say give it a bit. Always ask questions from them like their education, what they think of this or that, how they would handle certain generalized cases and so on. My therapist is always accusing me of analysing him because I know just about as much about him as he of me. I do therapy for depression and anxiety even though I know my slow forward progressing GID is responsible, I just haven't and can't open up about it yet. So I would say use an excuse to see a therapist for something other than GID. It can always come out later when they have gained your trust.
Also when asking things of them, ask them what the policies are of them sharing info with the military. Just insert the GID in along with a lot of other things like OCD, MPD and any other disorder you can think about or find on the internet. Yeah, I'm pretty good about beating around the bush. It's not a bad thing to be trusting believe me. Being non trusting has caused me way more problems in therapy than not.
If they act with an air of supeiority, request another theapist. If they act like it's none of your business about their background, how they would handle certain situations and just refuse to talk to you as a fellow human being, request another therapist. They should make you as comfortable as possible, make you feel like your in a safe zone and relaxed enough to open up. If they get frustrated with you and your lack of "addressing the problem" request another therapist. They should have an extreme amount of patience and realize there are people out there like me that just can not open up in a short amount of time. BTW I had a therapist with all the negatives that I mentioned. She told me she could do nothing else for me without giving me enough time to feel comfortable. She dealt with GID too which is really what I needed and what I eventually would have worked toward if she could have put up with all the other BS that I spew out during therapy. The therapist I have now, while not as extensive a background, is a lot better, actually talks to me, no air of superiority, is very understanding and all around a better person. Before and after sessions we talk about other things unrelated to psychological issues. I am becoming quite comfortable and am able to hint at things about myself more than I ever could have the other one. He probably already knows but that is a leap that I have to take myself without a push.
You should be able to tell with the asking questions, personality, disclosure policies and background in about three or four sessions. But all of that depends upon you and your own willingness to open up. Like I said, with me I am just now beginning to hint at my true self and it's been over a year.