I was in exactly the same situation almost. I'm pregnant and my partner didn't want his ex to find out about me and baby because he was scared she would stop access to his kids. We are living together and since she's found out about me and baby she won't let him have the kids anywhere other than her house incase he has them around me. I believe the reason is control, although other people on MN wondered whether there was more to it (I posted on here before). We have a whole new set of problems now, made worse by him keeping everything a secret. His problems have become my problems and the whole thing is a bloody nightmare that I could have done without and wish I avoided.
There could be any number of reasons he doesn't want his ex to find out. It may be to protect the children, they are old enough to understand and may become upset at the thought of dad with a new girlfriend. It may be to avoid drama with the ex wife, she might be the type to cause you both issues if she takes umbrage with the fact he's moved on.
It's also possible he and the ex wife are still emotionally and or physically involved and he doesn't want to burn his bridges, although hopefully it's not the latter (I couldn't bare the thought of that in my case)
Please don't be a push over like I was. Tell him you are not prepared to be a secret, that you are worth more than that. Because you are. We all are.
Sit him down and have a frank discussion, tell him you want to know exactly why he's so concerned about his ex finding out he has moved on. Tell him he needs to be honest with you now because sneaking around as though you are having an affair is a deal breaker - and mean it.
Hindsight is a wonderful thing and if I'm honest with myself, if I knew then what I know now about how my DP bends over backwards to prioritise his exes feelings over my own despite knowing it makes me unhappy.. I would run for the bloody hills. Not so easy now we have a baby due in 3 weeks and a joint tenancy which I'm bonded to.
Please whatever you do, don't become me!
I wish you the best of luck. You deserve somebody who's proud to announce you as their SO, not somebody who's afraid to cut the apron ties.
All the best x