I think you are being sensible wasp to be bearing in mind your age re:children. But I do think it is worthwhile trying to sort your thoughts out into realistic v not so important concerns.
For example - I don't think a self imposed deadline of 37 is useful (unless it has a real significance I've missed). But I do think that in your mid 30s you don't want to be left in the position of 'we'll try for a baby when I feel ready and I won't commit to when that is' from your partner.
(Not that that's what he's said! and I'm not advocating going at it like a bull in a china shop and giving him deadlines right now etc - esp as you have changed from being a person who did not want children to one who does. And he started a relationship with you when you were of your former view' AND you haven't been together very long)
I am biased because I have a lot of female friends who were strung along by their male partners with 'not yet, not yet... yeah, but not yet.....' in terms of children, and who hung on until their very late 30s or early 40s.... only for that partner when finally pushed to say 'no, I don't want kids' - or for them to run into fertility problems.
I am also a mother who had her 1st ds at 38 and is expecting another at 41, and that seems to be working out well - but I didn't fall pregnant at the drop of a hat, and not getting pregnant was a time of huge anxiety for me.
I don't want to be alarmist, at all - but one thing that shocked me as a naive mid 30 something was the whole 'waiting list' and eligibility issues for fertility treatment on the nhs. Here, the cut off age for ivf is 39 and a half. But there is a year's waiting list - and just to see the consultant who could put you in the waiting list was 6 months to a year. Add in time for tests and appointments etc before being judged a suitable candidate for ivf - plus the fact that gp's won't refer you until you've been trying for a year.... do the maths, and you realise that you'd need to be entering into that whole system not much older than 36.
There. that does sound alarmist, sorry. You have no reason at all to think you have fertility problems, I take it.
So what I would suggest is not that you sit and brood and fret - but that you get your thoughts clear and give yourself a few deadlines. Unless your dp brings the subject up voluntarily, why not leave the subject alone before raising it again in another 3 months. At a good time, obvs - not when he's stressing about money for example.
See what response you get then (and also if you still feel the same!). If he doesn't want to talk about it, ask if you can talk about it in a few months time. If he gives you a 'yes but not now' answer, try and get him to say what it is that would need to be different in your lives for him to say 'ok, let's start trying.'
i think those answers are really revealing. They need to be quite concrete and realistic - not 'When we have a lot more money' for example. Money concerns are sensible and practical - but only when they boil down to 'how would we both earn and what would be bring in', not 'we'll only have a child when I feel totally happy about money and have no worries'.
It is hard, but you have to be fair to him and what he wants, without leaving yourself waiting for a green light that never comes.