One of the challenges with your original clincher rims could be that they don't have a hooked bead which is used on modern tires to secure the tires for higher pressures. Can you take a picture of the inside of the rim with the tire removed?
You mention replacing the freewheel without specifics, such as needing to change the gearing, or are you trying to run a 6 speed instead of a 5? What do you hope to achieve? What are you concerned about in regard to your current freewheel(s)?
BTW, even if your current tubular wheels are marked 27" they are 700c. Since they fit with your current brake calipers you can replace the tubular rims with 700c clincher rims, which opens up 1000s of tire choices. If you were to stay with new 27" clincher rims, your tire choices would be in the dozens, so keep this in mind.
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Bob
Enjoying the GA coast all year long!
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