Morris Minor Owners Club

Googled it and quite a few suppliers of copper sealed pop rivets came up . Sometime misleading as although they claim to be stockists —— they are not !!!!!

Is this the sort of thing you're after? - Easily available on Flea Bay
I remember using these at school. You use an anvil (piece of bar)that has a dome machined out to fit neatly over the head of the rivet, this is placed in a vice and the parts assembled over the rivet and placed in the anvil and then the other end of the rivet is peened over - if you use another 'anvil' you'd get another domed head.
You can also buy hollow rivets that I haven't used but I imagine you use a centre punch to peen them over.

You won't get a 'Domed' look with Pop Rivets, simply due to the way they work. Only a Solid Rivet, with use of the appropriate reaction tooling with give a dome head.
A sealed Pop Rivet is one where the end is closed off, rather than having the pulled part exposed. This stops leakage through the rivet and ensures the broken off section of the pulled pin can't fall back inside.

Piece of steel with a start of a drilled hole in it to support the rivet head, clamp in vice.

Drift with a hollow point, much like a nail punch on the other side, and whack it hard:
You aren't trying to hold the Titanic together, so very likely that will do fine.

Decide what size rivets you're going to use, then get a tool with the right size dome.
Search fleabay for "round rivet snap head setting tool" and you see the punch tools are available in a lot of different sizes.

This one for example, MOD Surplus, unused, still in package. 5/32 rivet.
https://www.ebay.co.uk/itm/293678846360

Tools and copper rivets ordered, I've gone for 3/16ths (below), should be big enough for the rocker cover plates.
Thanks and best wishes,
Mike.
https://www.ebay.co.uk/itm/223682207998

There are lots of options with pop rivets - as long as that hole in the middle of the head is acceptable.

Rivet snapped-heads falling out need not be a problem for many jobs - if one can use a longer rivet than the minimum. One can fill the inner hole with silicone sealant and squeeze/bend the longer tail, to positively enclose the broken-off pin head and provide a good seal. Washers, behind the thin sheet, are useful to spread the load and provide a good seal

For normal rivets, the snaps are better used, because the rivet is expanded to tightly fit the hole as the rivet head is formed. We did not use snaps when replacing sections on reciprocating mowing machine knives back in the middle of the last century. One (should) soon learn how to hit the rivet square with a heavy hammer to expand the rivet shank, before forming the rivet head (or the section would loosen). It was a skill, like sharpening those sections such that they cut for a reasonable time without clogging the cutter bar. My Dad was an expert at that sort of thing.

I just use ordinary pop rivets and fill the depression in the rivet head with solder to get the ‘look’ right. Soldering best done on a hot engine. Not had any oil escaping.