If you stick with the 27"s then Panaracer Pasela are about the best tyre you can get. Rim-wise the choice is limited to what you can find.
Frankly, as 27" went obsolete about 25 years ago, I wouldn't bother going to the cost of new wheels in a very obsolete size.
For 700c I would you for Mavic Open Pro rims. 23c-25c tyres should work well, and a Campagnolo front hub which is nice and shiny.
For the rear, I would re-space the back of the bike from 126mm to 130mm, and fit a modern road hub. Shimano still make downtube shifters, which opens opportunities for something still looking right.
Remember that you brake callipers may not reach, so you'll need some others for the slightly smaller 700c wheels. I have some spare and barely used Alhonga deep drop callipers, PM me if interested.
However, you will then need more playing about if you continue with 6 speed. I would go with 9, simply because of the same obsolescence issues. At this point you may need a new chainset.
All this is building up to the point that once you start an upgrade path on a very old bike you can get into a spiral. Think carefully as to what you want - a bike to ride or specifically to update the Claud. If it's just a rider, then have a look as to what modern stuff is on offer. You should be able to break the Claud for bits and get a couple of hundred for it with a little luck.
Ride it or milk it!