When I was in homeport, they had UCT-1 down trying to get people to try out for it. I'm in Battalion and they came through and did a presentation encouraging people to do it.
You might need to talk to your recruiter and see if you could get it into your contract(not sure if you can that). You can always go through boot camp and A-School, get to Battalion, and then put in a 1306 (personal action request) to get orders to try out. Again, ask your recruiter.
From what the OF-13 rates have told me (OF-13 = Seabee rates) they didn't get to pick their battalions. They were handed orders and said congratulations.
For UCT you must be in a battalion or other CB unit for at least 2 years before applying.
Your orders are based on the needs of the Navy (i.e. were do they need your rate) and your dream sheet. Be realistic with your request - in other words if you want east coast ask for sea and shore duty in Gulport and overeas on the east coast (i.e. PR, cuba, spain, italy, etc). This informs the detailers that you want east coast. If you mix and match coast the detailers get the impression you really don't care.
The Navy may allow a waiver if you are joining with construction experience.
My personal opinion is the NMCB is where SEABEES belong, it is were you will perfect your rate, advance quicker, learn your role as a combat construction member and develop life lasting comadrie. UCT is specialized and is a adventure, it all depends on what you want out of your time in service.
