I know that (time changed) Lévy processes are actively researched in the academic world, including tools such as minimal entropy martingale pricing measures and fast Fourier transforms. To what extent are such topics used in financial engineering, i.e. in trading desks of options market makers?
Levy processes are not used for pricing derivatives and are useless in practice. When the task at hand is to price a derivative, i.e., working in the risk neutral measure, then using Levy processes is worse than useless, it is dangerous and should actively be avoided. You can add entropy risk measures, FFTs and other (practically) useless concepts from academia to that list.
If implemented properly, and there is a big emphasis on "properly", then they may be useful for risk management, i.e., working in the statistical measure. I would not use them for risk management either, but they could be used in this context for benchmarking purposes.