In my Peugeot bike restoration project, I removed the Nervar cottered cranks to be able to service the bottom bracket.

Because removing these cotters was a true pain (I had to drill a pin out :-( ), I decided to replace them with a square tapered bottom bracket and crankset.
The old fixed cup is left hand threaded, which is a good thing as far as I have read. (Since it's a bike from the 80's, I don't think it's using the french *strange
threading..., or does it?)

Here's a picture of the old fixed cup with a thread gauge, it fits the 1mm thread perfectly.

And here's me screwing it in very smoothly.
https://streamable.com/uuv5h

I got a BB-UN55 (68, BC1.37x24 D-NL) sealed bottom bracket replacement. Screwing it in starts fine, but the last 3mm won't screw in.. (Even using a bottom bracket tool, it still doesn't budge...)
https://streamable.com/ipzvb

What could be the issue here? Should I have my bottom bracket tapped? I also checked the bottom bracket thread with the gauge, and perhaps it doesn't line up perfectly... Should I buy another bottom bracket?

Is there anything sticking up inside the BB shell, a weld, chain stay end, a screw, etc.? I had this problem on an old bike I was switching to a cartridge BB, turned out the cartridge housing was hitting the end of one of the stays. Gave it a little kiss with an extended bit on my Roto-Zip and then it screwed in all the way, it didn't take much either. I've also had them hit on under the BB shell cable guide screw if the screws were a bit long.

French and English bottom bracket threads are mighty close.

English/ISO: 1.370" X 24 tpi or 1.375" X 24 tpi
French: 35 mm X 1mm (25.4 tpi)

Your measuring indicates you probably have FRENCH.

If you wish to go with square taper, hunt around on E-Bay, or....

It appears as if Velo Orange sells French bottom brackets.
IRD also sells French cups for their bottom brackets.

Edit:
Sorry, if the right cup is left hand threaded, then SWISS?
That may limit you to E-Bay or IRD.

Are any of your cups marked?

No markings on the cups, just checked. You are probably right, they are Swiss threaded.
This makes it a lot harder to convert to square tapered

Look at IRD.

One thing about the IRD bottom brackets is I think they use generic bearings, so they should be rebuildable.

If you can find a known French bottom bracket, you should be able to confirm threading on the left side.

Harder, but not impossible: https://www.ebay.com/itm/Japanese-Bi...8AAOSwvwhdQLkF

What is IRD?

NOT SHIMANO!!!

Interloc Racing Design

Italian, French and Swiss Bottom Bracket Conversion Cups ? Interloc Racing Design / IRD

Bottom Brackets ? Interloc Racing Design / IRD

It looks like their "Quad" is the basic BB they've had out for a while.
Quad Bottom Brackets ? Interloc Racing Design / IRD

The Defiant is the updated version.
Defiant Bottom Bracket ? Interloc Racing Design / IRD

Both take the French, Italian, and Swiss cups, and should have replaceable sealed cartridge bearings.

I believe Phil Wood also sells Swiss bottom brackets.

1mm thread spec is not the same as 24tpi thread spec. A Peugeot frame with a cottered crank and 1mm thread is almost certainly "French" thread, and an English/ISO bottom bracket will not work. You can either source a French thread bottom bracket on the used market, or from Velo-Orange or Phil Wood, or use a threadless cartridge bottom bracket. Exactly what will work for you will depend on the crank you intend to use when you build the bike.

He said the fixed cup was left hand threaded so it's probably Swiss.

...if your BB really is Swiss threaded ( and I don't know, I'm not in the room with it, but I've never actually seen a Peugeot that was Swiss threaded ). you can buy a cartridge BB in the correct length for your crank with plastic both sides, in the standard English threading and basically force the plastic retaining rings into your BB. The plastic will deform and pretty much lock the cartridge in place if you do this slowly and carefully.

It's much more likely you have French threading in that BB shell, and given that you'd be better off with one of those VO units in the correct spindle length.

The forcing of plastic (nylon ?) retaining rings into the BB is not best mechanical practice, but it works, mostly.

Keep the cottered crank. Just install the new pins properly with a cotter press and you‘ll never have any trouble removing them again.

Investing in a press will make future pin-removal much easier also. OP might want to keep a few extra pins on hand "just in case"; I've had stubborn pins' threaded portions bend even with the use of the proper tool to remove them.

I'll first try getting hold of a square tapered bottom bracket spindle. According to Sheldon brown I should look for a 3NN axle.
Doesn't seem that easy to find though. I'll go to the local *bike kitchen* to see if they would have some lying around. (Fingers crossed).

According to my imperfect eyes, that thread pitch gauge does NOT fit well onto the threads. I think you have a 24 tpi cup, so you need English threads.

...which photo are you looking at ? There are two.

Both, and the problem is clearer in the second. I could be wrong, but I think I see gaps.

Whew. It looks pretty obvious that it's a swiss BB given the pics and the videos. I might have square taper spindle in the 'box'. (What's in the box, what's in the box??)

If the fixed cup is LH, that is SWISS thread.

+ 1. This is the most straightforward solution.

It may be possible to run a cotterless spindle using the existing cups. That requires some trial and error but I did that to a 70s era Peugeot UE 8; so far it's working just fine.