In earlier tape recorders, how was tape spool angular velocity adjusted?
The system was mechanical.



Figure 1. The tape head and drive system. Image source.

Beneath the tape head cover on your photo is the capstan drive. A pinch roller presses the tape onto the capstan which runs at constant speed. (This is the critical constant speed control which insures accurate pitch control through the length of the tape.) The unwind reel is gently retarded by a brake and the take-up spool is driven by a mechanical clutch. (Obviously) the take up spool gearing has to be set to run quickly enough to take up the fed tape while at minimum reel diameter. It will slow down as the spool fills up.

For fast forward or rewind the speed will vary with the driven reel diameter.

All of the tape motion was driven by one constant-speed motor. The capstan was driven at constant speed by the motor. The various functions of the spools were controlled by engaging and disengaging drive wheels and belts. Clutches could be anything from simple felt disc between two sprung plastic clutch plates to fluid clutches, etc., depending on the quality of the machine. Some used 'dancer' arms to control the clutch to maintain constant tape tension with varying reel diameter.