There's two totally separate issues here.
First, deal with the situation with your child's father. Wait until you get the disclosure back from your request before you decide what to do next. Discuss your concerns with the safeguarding officer who contacts you. Make sure that contact is formalised in some way via the courts. This way there will be oversight if problems arise.
In my personal experience with Sarah's Law, though, ultimately the responsibility to keep your child safe falls with you, and you alone. I put in a request on someone who I now know to be a convicted child sexual abuser (he had served 3 years in prison prior to my request). The police actually wouldn't disclose this to me, citing 'privacy laws', but they said that if I suspected anything, it was my job to safeguard my child and I should act under the assumption that this person was a risk, since they would not make a disclosure.
Beyond that, I wouldn't leave my 6 month old child with anyone. For a few hours with a trusted, registered childcare provider during the day, yes. Overnight, no. Not with family. I'm not sure I would have left them overnight alone just with dh at that age. My dc were 2-3 before they spent any time overnight with anyone who wasn't us. Even then, it was in our home, where we knew it was a safe, controlled environment. And was with someone they know well.
Ultimately, trust your gut, but also recognise that if your child's father is bad news. If his family still has a relationship with him, they are either also bad news, or they are apologists for that sort of behaviour. If he truly has done something to harm children, my experience is that family who still have relationships are very keen to sweep it all under the rug, even at the risk of other children being harmed.