OP, nobody is judging unmarried couples with kids when that's what both parties want.
But that's not your case. That's what he wants: kids with no marriage, before living together, and on top of it he's telling you to do it ASAP because you are getting old! He doesn't care what you want, only what he wants. Otherwise he wouldn't push it and make you feel bad about yourself that you are getting old!
Money for his house, without any legal security for you and on top of it he decided his mother will live next to you! Again he wants to take financial advantage of you without offering you any kind of legal security. It would never cross my mind to ask anyone for money for my house, as long as legally it's only my house, not our house.
"“He’s worried for me that I’ll miss my chance” - surely that’s an emotion I should own?" - you are right, it should be your own emotion!
"thinking about it and now wants to marry me because he cares for me, knows how important it is to me and is willing to compromise." - but he only talked about it, he didn't do any actions. Also, from the start he's talking about "compromise" like he's doing you a favour by asking you to marry him. He even didn't ask you to marry him, but already is telling you that you should be grateful for it. That's a big red flag!
"I asked if he could give me a timeline and he said 3 - 4 years but he wants children first. I responded suspiciously and it ended up in another argument." - You told him how you feel, instead of talking about, he dismissed it and started an argument. That's what abusive men do when you don't agree with what they want. They start arguments to intimidate you, next time you'll think twice before talking because now you know it may end up in an argument. A big red flag.
"The pressure is hitting hard. I feel so overwhelmed by it that I can’t even think straight about how I feel because I’m thinking about how he feels. " - because you feel something is terribly wrong and you can't understand what. I think it's because you're again in a DV relationship.
"He has never seen this as a problem though and thinks it’ll work if I get pregnant now and eventually move in. This just seems absurd to me." - you are right, and it seems absurd to me too. If it seems absurd to you, then it's your right not to accept it.
"This is all combined with me moving into a house he is building." - you don't feel it's a house you are building together, you feel it's a house he is building. At the same time he expects 50% of the money for his house. In return, I assume he didn't mention about legally making you the owner of the house along himself once you start investing the money, right? Is he telling you that now you should give him the money for the house, but legally he will make you the owner one day, when he'll marry him? If that's the case, he's lying and is only after your money.
"I definitely want to have children - but the timing right now seems wrong with the other things grinding away at my consciousness (house/marriage etc)." - of course, it seems wrong, you don't own a house, you're not married, you even didn't live together with him. You want to do the right thing before getting pregnant and taking one step at a time, but he's telling you that you're unreasonable. You doubt yourself only because of him, not because you don't know what's right for you.
"I rent at the moment but was considering buying my own house. His response to that is that he wants us to be a team." - He already made a decision on his own about his mother living next to you for the rest of your life. He's stopping you from buying your own property by making you to invest money in his house. You may regret it for the rest of your life that instead buying a property for yourself you'll invest all your money in his house without any kind of legal security.
OP how are you now with money? Are you paying all 50/50 or according to the income?
What about pregnancy and maternity leave? Is he ready to pay for it or he expects you to pay 50/50? What about your future finances? Is he expecting separate budgets or not? What about the baby, who will pay for all the clothes, toys, nursery, school etc? Is his mother even aware that he's expecting free childcare from her? Maybe she doesn't know about it and isn't into free childcare? Maybe she'll expect to be paid for?
"He isn’t abusive to me like my previous partner. He is kind to our neighbours, helpful in terms of cooking/asking about my day/helping out with general things - he isn’t very emotionally expressive but he is genuinely a nice and calm person. " - OP, what you described here is the least you should expect from your partner. Because in return you do the same: you're not abusive towards him, you're kind with neighbours, you're asking about his day, you're cooking, you're cleaning, etc. What else do you love him for?
Ask yourself are you afraid of him? Are you afraid of telling him that you don’t want to do something? Start talking with him about all the difficult conversations about finances, disagree where you want to disagree. See if every time it will end up with arguments instead of a conversion.
Talk with your therapist about your relationship, I hope it will help you. 💐💐💐