I'm green with envy.
OK those synchro motors and the pear-shaped gearboxes are a standard "industrial timer" form-factor used to drive cams for simple things like 1970's washing machines. Once upon a time there were steppers with gearboxes made in the same form-factor - I came across them in my first job decades ago, though I don't think you will find one now - they are long gone, killed off by digital stuff and the NEMA standard.
You have a mount that is well worth keeping and upgrading to stepper control. What I would do is:
1. From the number of teeth on the worm wheel, calculate the nominal period of the worm for sidereal tracking. For dec, I'm guessing this has a tangent-arm - measure the radius and the screw-thread teeth and again work out a gearbox ratio to give a similar slew rate in terms of arc-secs/pulse.
2. Choose a NEMA stepping motor and gearbox (eg from pololu) to track at sidereal rate at something like 15 steps/sec, it doesn't have to be an exact match for sidereal - and design whatever metalwork is needed to connect gearbox to worm - this will have to be custom made, but done properly this does not devalue your mount. This will be capable of high-speed slewing and full GOTO.
3. Buy an OnStep kit to drive the motors, assemble and configure that. Some wiring and plugs & sockets involved but all feasible for a beginner if you can use a multimeter to check your connections, and are handy with a soldering-iron or crimp tools.
NB I looked at doing this to an old Losmandy G11 that had issues with the Gemini electronics but in the end decided I don't have the time for DIY, and replaced it with a CQ350.
Yep, did the math 
https://www.cloudyni...eq-clock-drive/
A regular (i.e. solar) day is 24 hours, or 86400 seconds. This gives an angular rate of 360 deg / 86400 sec, or 0.0041666666667 deg/sec, which is one degree every 240 seconds, or one degree every four minutes.
A sidereal day is approximately 23h 56m 4.0905s, or 86144.0905 seconds. The angular rate for a sidereal day is 0.0041780746238 deg/sec, or 1.0027379097096 times faster than the regular solar day rate of one degree every four minutes.
A clock drive turns the telescope's polar axis at this rate to match the apparent rotation speed of the night sky as the Earth goes around the Sun, which is 1.0027379097096 times faster than a regular clock would turn.
Various gear ratios are used to try and match this rate. One of the most common is to use a 359 tooth worm gear with a 4:1 gear combo attached to a 1 rpm clock motor. This gives a drive rate of
360 deg * 1/359 * 1/240 sec = 360 deg * 1/359 * 1/4 * 1/60sec = 0.0041782729805 deg/sec, with an accuracy of 4.74756267E-5 compared to the sidereal angular rate of 0.0041780746238 deg/sec.
However, large 359 tooth worm gears are getting very difficult to find – and are very expensive if you can find them.
360 tooth worm gears are available, but you need more intermediate gears to get the correct rate with a 1.0027379097096 overdrive.
There are several intermediate gear ratios are available (see attached classic paper from 1934) that give this overdrive rate.
The most accurate of these uses two gear combinations – 51/49 and 79/82. This gives a drive rate of
360 deg * 1/360 * 79/82 * 51/49 * 1/240 sec = 0.0041780736685 deg/sec, which gives an accuracy of 2.28645573905E-7.
I've found good sources for stepper motors and stepper motor gearboxes (not cheap either), so if for example I used a 60:1 gearbox on a NEMA 23 stepper motor with 200 steps per revolution, I'd have
360 deg * 1/360 * 79/82 * 51/49 * 1/60 * 1/4 sec = 360 deg * 1/360 * 79/82 * 51/49 * 1/60 * 1/200 steps * 50 steps per second = 0.0041780736685 deg/sec, but I still need sources for small 79, 82, 51, and 49 tooth gears.
Thanks to user Mercury-Atlas, found some of those old Hurst stepper motors that come in the same housing ( https://www.electron...er-12vdc-3-watt ). I ordered a couple of them, so we'll see if they fit.
What I need now is something to hook those surplus motors to see if they actually run. I can't find any info about that specific motor, did find some general information on the old Hurst web page ( https://www.hurst-mo...sabsgeared.html ), so all I know now is that it has a 20:1 gear reduction and maybe a 7.5° step (48 steps per rev).
My guess from looking at the intermediate gears inside the mount casing is that it's a 2:1 ratio, so that means that the 1/240 rate to turn the 359 worm gear would be 1/2 * 1/10 gear reduction * 1/12 sec = 1/240 if the motor is 5 RPM.
If these old surplus steppers are geared down to 20:1 as advertised, then the setup would be 1/2 * 1/20 gear reduction * 1/6 sec = 1/240, so I'd have to run the motor at 10 RPM, or eight steps per second ( 0.125 step /sec, one step every 125 milliseconds) for the sidereal rate, sixteen steps a second for 2x, twenty-four steps per second for 3X speed, etc.
