Discrete transistor circuits typically do not create the delay effect, but they will be found in delay effects performing critical functions.
A discrete transistor delay effect is possible. For very short delays, really long chains of passive all-pass filters can provide enough phase shift to constitute an actual delay. Historically this was done, but the concept was probably dead by the 1970s, if not earlier.
A discrete transistor sample and hold circuit can be constructed, and a long chain of them could create a discrete bucket brigade delay. This would be a massive construction job, and keeping capacitive leakage under control would probably be difficult (i.e. your delays would be very dark sounding). Historically, early sample and hold and bucket brigade experiments were probably done discretely, but the microchip/IC revolution really made the idea for audio delay a practical thing.
Discrete transistor filters for BBD circuits are typical of Boss effects and others.
Boss even used a discrete transistor clock for the CE-1 Chorus Ensemble BBD chip.
Tape machines have to bias the tape with a constant tone at around 40-50kHz while recording. This bias tone is then removed before playback (note that the tone is also supersonic, but it can cause audible artifacts it not filtered by a "bias trap" before playback).
All tape delay effects will have a bias oscillator. Many are constructed with discrete transistors.