If ATC tell you to standby and don't get back to you, then you are entitled to think that they have forgotten about you.
What you need to do is give a gentle reminder.
However, "Solent Radar G-FA request" is not in the R/T manual and gives them no idea of what you want.
"Solent Radar G-FA request CTA crossing VFR Altitude 4000ft" while not being book perfect does give them a clue that you are not one of the 100s of flights that are going no where near the airspace but call anyway.
Even better, you could also ask Farnborough (the LARS unit for the area that abutts the Solent CTA) on initial call that you will require a transit of the CTA so that they can co-ordinate well in advance with Solent.
Did you know that you would require the transit pre-flight? If so did you file a plan or telephone the request to Solent yourself?
If ATC say "don't call us, we will call you" and fail to call you, they have forgotten about you or are ignoring you. ATC would not stand for a pilot saying that to them and would file an MOR. You can do the same if you feel the situation was worth highlighting.
Having said that, you could have moved just few miles east and stayed with Farnborough at 4000ft before talking to Bournemouth who are the next LARS unit for the low level transit below the Solent CTA 2000ft base. All outside controlled airspace with only an extra 2 or 3 minutes for the diversion.
Solent cause their own problems. They should not talk to aircraft who are not routing through their airspace. Every flight that calls them with no intention of routing via the CTA or CTR should be immediately told that FIS is available from London on 124.75. Once pilots get the idea, the frequency will quieten down a bit and those that do require a transit can be assured of getting their message through. Unfortunately, Solent talk to lots of VFR OCAS flights while they are quiet but then dump everything when the airways arrivals start pouring in. The simple answer must be not to get involved in the first place.
Regards,
DFC