I'm probably an outlier but I think a month is quite soon to expect commitment. For me, I'd have only seen them a maximum of six times in that month, probably less. And those of you with kids, even less.
So, if you've had the chat and it's gone well so far, I'd let some more time pass. Re the tests, he won't have forgotten, every time he puts a condom on he's reminded, but something is stopping him. Not interested in tedious admin, scared of what it involves, too lazy to make the appointment, no time to go to an appointment, etc. (Many places you can just order a test online to do at home, does he know this?).
But worst case is he thinks you will forget and he won't have to do it. Which is disrespectful, obviously.
So, in your shoes, I'd probably leave it another month. If things are going well just let that lay there as it is for a while. In the grand scheme of things, two months is nothing. But check how he gets on with those tests - my long term ex never did them, he just bluffed about it until I stopped asking. It was a red flag, I should have paid more attention.