So a weird thing about human nature is that if we are given a say in something, even a heads up about it, we feel more in control and happier - even if we don't do anything differently as a result. We don't need to be allowed to make the decision, just kept in the loop.
Your endgame has got to be an ongoing friendly relationship with the person you live near - if it's awkward it will feel slightly tense all the time, you'll be looking over your shoulder in the co-op and so on.
If you don't tell your relative until it's a done deal, you risk them thinking you've been a bit cold and you haven't let them into a nice thing about your life. That makes them feel a bit exposed- even, that you don't like them that much?- and sets the whole thing off on the wrong foot.
This is a great opportunity to let them in a bit. Asking for help makes people empathise with you more. Yes, it's a new estate and they don't know the neighbours, but are they in the house already?
Pop them a text saying "Need to ask you a favour! I've had the chance to put an offer in on no 14 Your Street. I'm quite excited about it. Can I come round sometime and find out about what the purchase was like from your perspective- was the snagging all done OK, any quirks about the houses I should know about?" Then they probably say oh right yes, then you reply and say "It's not a done deal yet - would you mind not mentioning it to Mum and Auntie Sandra, they get a bit flappy with me and would ring me all the time asking about it! amused emoji, etc etc".
They get the chance to help you and tell you how moving in was from their point of view - so they see you as a positive thing, a new neighbour coming into their area. And they're on your side and less likely to tell Mum all the details. In fact, this is one of those moments of life where you can move to a different adult stage and break some traditional patterns ("but everyone tells everyone everything!"). You and cousin are allowed to have a slightly different thing you both know about for a while.
If you have the attitude of I should be able to keep my business to myself - I've lived in this town all my life - they are more the incomers than me etc etc. - it will come across. Even though you are TECHNICALLY right, you are wrong RELATIONALLY.
Weirdly if you like to be private, giving a little information in a breezy way works better than being like an oyster and not saying anything. People just ask a few questions then forget it, whereas if they think you're keeping something from them they get all huffy.