This is an update to my Original Post.
Was at LGW lounge yesterday on my outward journey to Cyprus with a 6 hr layover in BA's lounge. My planned 13-hr layover in London is for next week on my inward journey.
So I asked BA's GC desk in the LGW lounge about the possibility of getting into the lounge through flight connections on my return for a quick shower and then go land side for sightseeing, she said it is perfectly doable but I will have to be escorted by security officers from the lounge to the immigration. They will arrange the escort. So seems like good news.
I will report back what actually happens on my return.