Replacement for 50+ year old solder fuse technology
I don't know if it will be possible to determine at what current the solder fuse &quot;blows.&quot; But as a practical matter, you could measure the normal current consumed by the clock, double it, then round up to the nearest standard value fuse.
IMPORTANT: fuses are sized to protect the wire, not the device at the end of the wire. Wire and fuse must be chosen together. And of course, both must be large enough to support the load. Once you fuse the wire, it is OK to use heavier wire. The fuse imposes a minimum size on the wire, not a maximum.
Based on the comment, maybe a thermal fuse would be more appropriate.
It can be tricky to try to protect against a stuck solenoid using a normal over-current fuse. The reason is that you WANT the fuse to blow when the normal operating current happens, but only if it happens for a long time. Unfortunately, if the solenoid gets stuck on, the copper wires will heat up, increasing resistance and reducing the current. This means you need a smallish fuse. But if the fuse is too small it may blow when there is no fault. On the other hand, if it is too big, it may not blow even when there is a fault, because the solenoid current will decline over time. It is kind of like chasing a ball downhill. You try to catch up with it but it goes faster and then you speed up, but by then it speeds up too, and you never catch it. I have been down this road before trying to find the goldilocks fuse.
If the solenoid is accessible it may be better to put a thermal fuse in physical contact (with silicon or something) with it. So the solenoid current goes through a thermal fuse (AKA thermal cutoff or TCO) which physically touches the solenoid. Then if the points stick, and the solenoid starts to heat up, the heat will reliably blow the TCO. TCO's are subjected to a lot of regulatory stuff and are very reliable because of it. Protecting against over-temperature is their one and only job.
There are wide ranges of temperatures available.
See just as an example the NTE8070. (No affiliation or commercial interest on my part and there are other parts out there this is just an example).
https://www.nteinc.com/specs/8000to8999/pdf/TCOs.pdf