This was me 5.5 years ago. I've had a lot of abdominal surgery, and midwives were struggling to diagnose DD's lie due to scarring; I was sent for a scan at 36 weeks, which revealed she was transverse. I did have ECV (at about 37 weeks) which was initially successful, but DD turned back. I was admitted at 37+5, and had a planned c-section at 39+1, with a tedious 10 days in between - I was in hospital, but allowed anywhere on the grounds. Incidentally, the 50% success rate for ECV is for breech babies; my hospital didn't know the rate for transverse ones.
Once in hospital, DD's position was checked by hand twice a day; she managed every possible position (including head down), though transverse was her preferred option. I was initially told that if she turned herself (and went head down) I could go home to wait for labour naturally. However once it became clear how unstable she was, I was told she'd have to be head down for 48 hours before they'd consider her stable; she never achieved that.
I was also told that a transverse lie is very rare for a first baby, and they couldn't see why; DD was a normal shape, I wasn't carrying excess fluid and my uterus was normal. However I suspect that I carry scarring from previous abdominal surgery, and I've had bits taken out so I suspect that my internal organs probably aren't in quite their normal places anyway. We just have the one child, though the possibility of having another transverse baby wasn't what put me off - my c-section recovery was good (if not much fun) as the c-section itself had been done slowly and cautiously, which I'm sure helped.
Hope that's reassuring - any more questions, feel free to ask!