Coming on to say that the 'Nanny Dog' thing is a total myth. Coined in the US about pitbulls, now has made its way over here and been applied to staffies.
The bottom line is that if you are going to put your kid's safety first, then no, your PIL will NOT be looking after your baby in their home with a dog you know they won't bother to train.
Yes to PIL coming to yours.
No to PIL having baby at theirs where dog is. Flat no.
People who don't take responsibility for training their dogs are the liability. EVERY dog has is liable to turn and a baby is defenceless. No dog should be trusted with a baby or small child. The kind of people whose default setting is 'aww, it'll be fine, no need to train Fluffy, look how soppy he is' are exactly the kind of people who will also say, aww, no need to keep Fluffy in the kitchen when we have baby, he'd never hurt a fly, what DIL doesn't know won't hurt her.'
The breed isn't the point, although it's simple fact that the jaws of a staffie are a lot more powerful and dangerous than a lot of other breeds. However there have been babies killed by jack russells.
You need to politely put your foot down now. No, you are not putting the responsibility of baby+dog on your PIL, you are laying down a red line now that your baby won't be around a puppy without you there for a very, very long time.