It’s been a while, so forgive me if I’m out of date, but as far as I remember, ATC have no knowledge of an operator’s “runway slot time”. This is arranged directly with the Airport Authority by the aircraft operator. The trick is to merge this with any possible ATC slot constraints, which normally only apply to the departure airport.
I once arrived about 2 hours early on a ferry flight at LHR. We knew that we were way ahead of the allocated time but didn’t fancy hanging around at the airport of departure. ATC were happy to let us land but our Airline management lost their sense of humour when “Heathrow Airport Limited” raised a complaint.
So, the Captain will be made aware of any arrival slot times and will liaise with ATC at the airport of departure to achieve a take-off time that fits, subject to any other “flow control” measures that may be invoked by en-route ATC.
Hope this helps to explain what you experienced.
As to why Gatwick were so strict about slots, given the lack of traffic, I have no idea.