I may have this wrong but my understanding of the military use of 'overshoot' is that it refers only to the IFR context, i.e. you report missing the approach with 'G-ABCD overshoot', and if training you can 'Request ILS 19, overshoot to further' (= civilian 'request ILS 19 low approach and go-around, further approach'.
Go-around is used for a go-around into the visual circuit only.
However I may have this wrong, as it's terminology I use rarely.
Tim