A place to ask bicycle repair questions, and for bike shop monkeys to share advanced non commercial wrenching resources (no YouTube self promotion). This is only for repair related topics.
While 27" is an old standard, you can still get modern tires in that size. Bike Tires Direct has a decent selection from quality brands. (link) No need to replace the wheels.
I'm partial to Continental Gatorskins myself.
Your 700cx35 tube will work fine in a 27x1.25" tire.
The tube are good. If you wanted new wheels you'd need to check clearance with 700c wheels and make sure your brakes reach. I have schwalbe marathons on my 27"wheel bike and love them.
Thanks! I'm hesitant to replace the wheels because I have an internal hub on the rear and replacing the wheels would mean paying for a rebuild. But based on doing some more googling it looks like that would be my only realistic option down the road.
Edit: I just wanna say y'all are so lovely and helpful here.
I wouldn't (and didn't) switch my 27s out and have had no problems. Wait until your hubs need servicing and then revisit and see if there is anything a new wheel size would help you with.
Second this on the brake clearance checking. I had an old west German steel road bike with 27 x 1 1/4" wheels and thought I could just swap in modern 700c road bike wheels and be happy. However the brake calipers on the bike wouldn't work as the rims were now in a slightly different location and wouldn't adjust far enough to reach. That said you can get extenders and longer reach brake calipers if you're fully committed to putting a modern wheelset on, but worth checking everything will work/fit before going too far down that route.
So I linked below what is most likely not the best tires out there but they are a really good deal. A friend of mine recently was fixing up a bike of a similar age and used those them.
I only use Conti Gatorskins or Hardshells. I raced with other stuff in the past, but without racing, less flats is all I care about. Generally speaking, in my area, you can pay more up front for the Continental's, or you can pay the same amount or more in replacement tubes due to flats with cheap tires. Be aware that some tires like Conti's are directionally oriented on the bike.
Pro commuter tip, buy tubes in bulk online so you always have a spare ready to go. You will find cheaper options buying 8-12 at a time from many online sources. Also if you can source a CO^2 inflator that works with threadless cartridges, get it. You can get threadless cartridges much cheaper online.