Do military aircraft such as the Hawk in question have flight data recorders or CVR's fitted to them?
Broadly yes, as far as newer types are concerned. The TMk1 has an ADR but no CVR. Fitment of a CVR has been recommended in several BoI/SI's but the MoD haven't yet taken the plunge. With the prospect of a 2030 oos date interest was being raised again, but with the latest defence review that now looks unlikely.