I also wonder if converting old coal-fired plants is possible. If the turbines, generators, and the power distribution stuff is in place and in a good shape, maybe setting up an alternative source of steam could be more economical than building the whole thing from scratch.

Typical coal plants use superheated steam at around 540C and turbines designed for that, whereas LWRs produce wet steam at 300C, so no.

High-temperature reactors can produce superheated steam, but then it's not a LWR and you give up on all the knowhow how to run those. The Chinese are planning to deploy gas cooled pebble bed reactors which could replace the coal furnace at existing plants (HTR-PM).

Alt. take: one could not reuse the turbines, at least not efficiently. One might be able to reuse other parts of the steam/condensate/feed systems. The electrical systems could largely be reused.

I doubt that reusing anything other than the power distribution and cooling loops would be economical.

Don't forget maintenance stations and other office housing required to run the plant.