Thomas,
If you file an IFR flight plan all of your clearances within the London FIR will keep you within controlled airspace. This applies equally to your arrival and departure from EGHI.
Don't forget that EGHI is PPR due to restricted parking - and they are not cheap at all. From my experience EGHH is by far the cheaper of the two.
Should you elect EGHH as your destination, again the clearances issued by ATC will ensure you remain within CAS and a known traffic environment.
Greetings from EBBR.
Oh dear!