I just looked at what I would file for that. Firstly, here is my stab at irrelevancy - there are no standard route departures to file from popham so that isn't a help. The otherwise excellent standard route doc would be very useful if you were departing from somewhere that was in it though...
NATS | AIS - Home
More helpfully - using the also excellent Autorouter - Two routes validate:
EGHP N0121F130 BIG Q70 KOK KOK7A EBBR
or
EGHP N0117F100 MID/N0117F100 IFR M140 DVR L9 KONAN L607 KOK KOK7A EBBR
I would probably file the second as I would guess the first is unlikely to result in a climb into controlled airspace much before Biggin.
My guess is that you would indeed get to fly as filed outbound, but possibly not on the return leg. Or vice versa. BUT. it really depends on factors such as the runway in use at both Heathrow and Gatwick - because of the traffic flows. I asked a similar question about Denham - Oostend in the ATC Issues forum and received some really useful answers there. I would think it unlikely that you'd be vectored overhead Gatwick, and if you do get vectors rather than direct MID, direct DVR it will be to avoid the prevailing traffic flow, especially if you are in the opposite direction to it. And probably a descent too, to avoid triggering TCAS alerts. Which in turn may drop you out of controlled airspace. And, yes,. likely well south. No easy way to predict. But for more details do ask the same question in ATC issues... some really helpful folks there.