How do the compressor stages keep spinning after start in a free turbine?
In a C20-250 engine the compressor section is driven by the high pressure turbine (Ng spool, not N1) independent of the power turbine.  The power turbine drives the gearbox and subsequently the output shaft via a concentric driveshaft to the gas core spool.

I believe you’re confusing the configuration of this engine by assuming that all free turbine turboshaft engines are laid out similar to a PWAC PT-6 engine.   While the PT-6 design certainly excels in terms of simplicity as well as modularity for ease of maintenance, it is not necessarily the only configuration for free turbine engine. Plenty of other free turbine turboshafts have other unorthodox layouts but all operate according to the same principles.

A cutaway of the C20-250 engine appears below.  Note the non-axisymmetric air channel carrying air from the last stage of the compressor section to the aftmost part of the engine for introduction into the single can-type combustor.