If you get away from BASIC commands and copy the tape routines up into RAM you can mess with the timing values, and with care you can get it from an average of 1800bps up to around IIRC 12000bps before things really fall apart. Probably with better interfacing you'd get even faster.
With more conservative values, this is how "speedloaders" worked. In the late 80s or early 90s one publisher released games on CD where it would load in the loader at normal rate then ramp up to some unholy speed that CD had the bandwidth and stability for but tape did not.
Used to play Duke Nukem 3D over a null modem serial cable.
In the days of the spectrum where you loaded via an analog audio input from a tape, and saved via the output, you could move from one to another in the same way, skipping the tape.
Apparently this was about 1kbit per second.
But if you were connecting two terminals directly with no modems, thus the modem was "null", you needed the cable to do the signal swap instead. Hence the name!