Hello! I'm so sorry I didn't see this earlier, I hope you still see my response!
I have left Bucharest now but lived there for 6 months and was pregnant there. In my experience it's a really baby-friendly city, with loads of parks and playgrounds.
We lived near the Arcul de Triomf, in the north. There is a huge park there called Herastrau (sorry, I'm skipping accents because too lazy :) ) which is always full of mums, dads and grandparents walking their babies in strollers, and has loads of playparks. There is also a big lake which helps to keep it cool in summer. There are other parks all over the city, but I really loved Herastrau. Dogs mostly stay on their leads (it's more forest with paths than open grass) but there are some fenced off areas where you can chuck a ball around to give them a proper run.
When we arrived it was winter, and in cold snaps it can be -20, which is obviously pretty tough with a little one as you can't get out and walk. For the most part, though, it wasn't that cold. Summer is stinking hot in Bucharest, so you will need aircon. If you have moved around a lot you will probably already have a handle on getting out in the mornings while it is cool. And the mountains to the north, Transylvania and the Black Sea beaches are all great to escape to.
I would go back to Bucharest in a heartbeat - you are going to have a great time! Younger Romanians generally speak excellent English and are very welcoming, so if you don't find any expat mums, I am sure you will be embraced by the Romanian mummies!
Let me know any questions! We also move around a lot, and I'm pregnant with my first, so I know what it's like.
Paws