There are flight students older than you and some of them fly jets. Age is not a factor at all to fly jets - just performance. I have recently seen flight students in the mid 30's but they are usually prior-enlisted. Flight students in their mid to late 20's is very common now a days. Typically you must start flight training by age 31. I'd call an Officer recruiter now to find out more about getting a flight school slot. Army, Navy, Air Force, Marines, Coast Guard may have an active duty job for you. Ask here or with recruiters about Air National Guard and AF Reserve opportunities.
Having a bachelor's meets the education prerequisites. If you pass a physical your timing may be very good. I wouldn't delay, you are still young but not getting any younger. Recently there have been posts by pilots going to Navy OCS and perhaps they can give you more insight. They are probably taking the fastest route to flight training and wings, in my opinion. I think these folks are coming in at a good time. Military spending is going to go up before it goes down again.
Good luck!
Mike