Sharp-eyed viewers may have noticed that the mech in the pic
ToniH. posted is shifting to a 34t freewheel. Don't expect to get that range from your Record just by putting a GS cage on it. Though it
might work, the one in the pic has been hybridized. The upper knuckle is a from a "Sport" (not Gran Sport), the single-pulley mech intended for 3-speed city bikes. That upper knuckle has a spring, so grafting (riveting) it onto the Record parallelogram arms makes a double-spring shifter, equivalent to the Simplex of that era. Vastly increases the range, even with the original Record pulley cage. I fact for the bike that's on, I went back to the Record cage, even though I'm using a wide-range triple in front (20t difference) and a 14-30 freewheel. The short cage has enough wrap to reach all 15 gears.
Here are pics of it handling the extremes, big-big and small-small:
I don't know what size freewheel you can get with the non-hybridized Record by swapping the cage to GS. Probably a little bigger than with the stock cage, but the main advantage will be letting you use a wider range of chainrings. The Record was intended for a narrow range up front, typically only 3 or 4 teeth, the half-step you get with a racing size freewheel of that era. The GS cage will allow you to use a triple or wide-range double.
Note I do
not recommend chopping up Records and Sports and Frankensteining them together. This was a "fun" exercise in some ways, but also a total PITA. And just about any touring mech will work better with much less cursing and excess stomach acid. Oh and for the record (no pun intended), both the "organ donors" were dead. The Record had a broken top knuckle, and the Sport had a twisted cage, so no working mechs were sacrificed.
FYI I also have a hybridized Gran Sport, the earlier bronze-steel version. The Sport upper knuckle fits those too.
Mark B