LHR T5 Departure - you need to be clear security by 1825 (hard cut off time, if you dont' make this you'll be offloaded from the flight) .. Working backwards from that the latest HEX train you could catch is the 1755 from Paddington (arrives T5 at 1816) .. this is do-able with hand luggage only, boarding pass in hand and no messing around ... if you need to check bags/check in you'd need to (at least) be on the train before that (1740 or 1725) ... Assuming you could catch the 1755 HEX train and pre-purchased your tickets then you could leave the Intercontinental around 1730 in a cab ( Question for the audience, whenever I get a cab to Paddington they go around a super long way via Edgware Rd. and the start of the A40 around the back of the station .. can you get a cabbie to drop you to Paddington a faster way? )

Piccadilly Line tube at rush hour ... I think theoretical journey time from Hyde Park Corner to T5 is around 45 mins ... but at rush hour I'm loath to trust that line much, only one in 3 trains goes to T5 and spacing can be very erratic and at that time of day the trains will be absolutely packed by the time they get to Hyde Park Corner .. I would want to be going into Hyde Park Corner station at the very latest 1715 to feel safe with that option

LCY - If your flight is also due to leave at 1900 there is no security conformance time, with hand luggage only, pre-checked in you could certainly turn up at LCY at 1830 or even 1840 and be safe enough .. Assuming you aim to arrive there at 1830 then working backwards from that it would take 10 mins on the DLR train from Canning town to the Airport and 20 mins on the Jubilee line from Green Park station to there - Green Park is a 5 min walk from the IC hotel .. Assuming the tube is running fairly normally (the Jubilee line is in my experience more reliable in this regard) then I'd be happy leaving the IC hotel at 1745 going via this route.

Kind of long-winded above, but I think my conclusion would be .. if you've no luggage LCY is the better option - you can leave slightly later and don't have the dreaded 35min security cut off hanging over you either (so you've a little more margin for error) If you do have luggage take a taxi to Paddington Station and HEX to LHR

Driving to either LCY or LHR all the way at that hour is a disaster and will take longer than any of the above methods - don't do it no matter how much the hotel concierge might try to encourage you into their car service - In the past Heathrow might have been do-able in 45-60mins from there but with the ongoing roadworks at the Hammersmith Flyover I doubt you could drive to Heathrow from Hyde Park corner in much less than 90 minutes at that time of day.