if I purchase a small Belfast sink fron England to undermount in a bathroom vanity and ship it to the US, will I be able to use standard American plumbing fixtures (drain, pipes, faucet including a pop up drain)? If not...are the proper plumbing fittings readily available in the U.S.?

Thank you David! When having this house renovated post purchase, much of the bathroom was gutted and the sink I had installed is one I bought new some years before and was waiting to use (American Standard Standard Collection Console sink with oval glass shelf.) Because the works underneath are totally exposed I bought the best chrome P-trap and fittings a local plumbing supply house had. In a month or so I'll have my plumber back to install a pump for my utility sink, and he will need to add another vent stack I believe, so I'll have him see if the sink is connected to the bath vent, and if the vent has any blockage. I'll ask him about the air admittance valve - thank you so much for that information!

Hopefully you have PVC or some similar plastic pipe. The proper thing to do is to pull the vanity, open the wall, raise the pipe, reinstall the vanity. The picture you've drawn begins to look like an S trap. These are not code compliant since they can lead to emptying the trap by siphoning.

What all isn't working? You might have to wander the internet for washers and things for the old faucets--or replacement handles at least. Does(did) it actually have a popup mechanism in the middle? Not just a thing to hang the plug off of? neat! I'm sure you can ignore it when you replace/redo the faucets if it polishes up OK or remove it is you can use the hole for a spout. If you go with a new faucet with a mixer, take out the old parts and measure the holes first. I had an old sink with tiny holes once so there was no way to get a modern faucet in. Then you'll know if a widespread (minus the popup) will work. As cute as 2 spouts and a thingy in the middle are, some of us like warm water. :) Is the glaze in the bowl crackled? It might be too far gone for heavy use but could still be very cute in a powder room.