First ever mechanical/automatic watch for a 12 year old

May I suggest a Seiko 5? It's small enough, cheap enough and nice to see at the wrist. There is a lot of the to choose from...

Example: SNXS77, just to mention one. Or a SNXS79K1(I own one).

probably seiko 5 snk series. or if you can get an alba it would be very affordable ans nice to, they are the subbrand of seiko also

In addition to Seiko 5, you might consider a Vostok Komandirskie, or look at Berny or Ochstin on AliExpress.

I’m going to get a ton of flack for this, I’d imagine, but…have you considered an Invicta Pro Diver?  Sometimes they can be found around that price point.

https://www.ashford.com/invicta-in-35848.html

Maybe a timex? Not sure if there’s a mechanical timex for that price though?

Mumbai special. Or just search Amazon or whomstever for the cheapest mechanical. I had a Winner brand that was like $25 at the time and was really not so bad. I can't remember the other Chinese names, but they exist.

An Invicta Pro Diver or a Vostok Amphibia. You can find Pro Divers on the Invicta website for as low as $40 vucks, and the Vostoks are available on amazon. 

Something durable would be my advice and with kids you never know when they will end up wet, so a diver would be a good choice. If you don’t mind a micro brand checkout Deep Blue. They have a great range, are well priced and are well built. I introduced my nefew‘s to their first automatics last Xmas and chose Deep Blue. I have one myself with a Selita movement, but most of them will have Seiko movements. You could do much worse.

Given you price range, I’d say, in line with others either a used Seiko 5 from eBay or, consider a quartz Timex. Something like a new Timex Expedition. While it is quartz, it’s a cool design and might go a long way in helping your cousin form a deep appreciation for watches. Then he can get a more expensive manual or automatic later. 

I’d say something inexpensive and somewhat smaller like a Seiko 5 or an Orient of some sort. 

I would go Seiko as well. I have and love Timex but at the lowest end they cannot compete with value for Seiko. 

My boy is 12 as well and I got him a Seiko 5 SNK809 earlier this year. The 37mm size fits him perfectly and he wears it all the time (but admittedly he does wear his GShock Rubiks Cube LE quite a bit as well).

One more vote for Seiko 5 here. Not that long ago, I have arranged a modded (sapphire instead of Hardlex) SNK809 "pilot" for my little half-brother as a communion gift. The kid's really happy with it, so I reckon I chose right and can recommend it. It's something a kid can wear and won't grow out of it, ever

SNK809 for sure.

Serious question: how vital is it that this watch actually provide reliable, accurate time? Normally I wouldn't ask, but if the intent is to get a child fascinated by the mechanical geegaws, and there are other ways to know the time everywhere he goes, this open low budget windows to all sorts of dicey used stuff, right?

First watch that came to mind was like everyone said, a Seiko 5. If not that I came across this. It's a Boyzhe. It's a nice looking watch. It feels of really good quality. It has a working tourbillion. It is an automatic and is really heavy duty. It feels bulletproof. It came in a bundle of watches that I bought but looking on Wish the other day I seen them. They were under $30 US. It's a lot of watch for the money. Just a thought. 

I normally wouldn't suggest Invicta, but their 8926OB diver may be a decent choice. It's essentially a Sub homage, but it offers an exhibition caseback, which may be something that helps further spark the interest in watches. I had picked one up to use for practicing movement regulation, and had been able to regulate it to a point where it was running +2 SPD. I picked mine up for $92 CAD in 2019 from Amazon. Their list price is ridiculous, but you should be able to find them online for close to your budget.

the intent is 100% to get him fascinated by the mechanical aspect of watches, he is already showing interest 

Roberto's HMT suggestion is spot on, even though I'm not aware of them having the de rigeur exhibition back. I might also say pocket watch, provided he seems the sort to be into that. Otherwise a Mumbai special Seiko 5 is great, as the price is low and he's not going to care about authenticity.

This is more expensive than 50 euros but is a good all around watch with a decent movement and whether or not I would actually buy it for a 12 year old would just depend on how generally responsible the young person is. But my pick is the Timex Navi XL.
Just recently their Red Bezel/Black dial was on sale for $150 USD on Timex's website but they 
arn't available there anymore so you have to get one from a reseller (example below).

https://www.amazon.com/Timex-Navi-41mm-Automatic-Brown/dp/B08K2SCY7B/ref=sr_1_1?gclid=CjwKCAjw3qGYBhBSEiwAcnTRLiQSEDZfsN0R-m9zEv7gOS6Ayly7MmtsUvYrrqBE9s8Z2G0DUchOTBoCaccQAvD_BwE&hvadid=557476346118&hvdev=c&hvlocphy=9028543&hvnetw=g&hvqmt=e&hvrand=5988857553695979684&hvtargid=kwd-826503068041&hydadcr=7470_13185762&keywords=timex+navi+xl+automatic&qid=1661563535&sr=8-1

A bit surprised no one has brought up Orient! Look at the Mako II. Reliable movement, he can get it wet all day long and he'll have something different than a Seiko 5 (no offense, I like them, just saying...).