Layover at London Heathrow

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I have a 4-5 hour layover at Heathrow en-route to Nairobi from New York. Will be flying Business class, so had originally planned to use this time to take a shower in the British Airways business class lounge and possibly get a short massage (they apparently offer them for free) before continuing on to Africa on the remaining 8.5 hour flight. I would prefer to go through customs, and go to the nearest airport hotel to use their fitness room to do 20-30 mins of cardio, followed by a shower, before returning to the terminal. I know that going through customs at Heathrow will be relatively quick because I'll be in Business and can go to the Fast Track line. Obviously have no luggage to retrieve becaue all will be checked through to Nairobi. My concerns/questions are: Do I have enough time? Will I be allowed to go through customs and leave the terminal building before returning back for the connecting flight? Any suggestions on which airport hotel to go to, if I decide to give this a try? Are there any fitness facilities in the BA terminal gate areas so I don't have to go through customs? What about fitness rooms in the terminal itself that I could use instead of going to a hotel?
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You could also try the BA arrivals lounge at T4, which you will have access to. I think they've got (admittely limited) gym facilities.

Alternatively, your most convenient would be the T4 Hilton, as it's attached to the terminal.
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I wouldn't count on getting a fast track as customs doesn't have a fast-track (they don't check much anyway ) and the BA Fast-track immigration line is only open at certain times.

I don't see any problems with getting through to landside between your flights and it appears this is the only way you'll get to a gym! The arrivals lounges which are airside don't have gym facilities although I believe the T4 Hilton (walkway from the terminal) does have one. Never been to verify though.

If I were in your shoes (or should it be snickers) I'd avoid landside distractions, go to the Molton Brown Spa in Departures area, get a shower and massage booked up and spend the hours in between going up and down the stairs in the Lounge Pavilion You have three floors to contend with. I guess it depends how keen you are to get to a gym.
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Sorry, two points of clarification: No gym facilities in the BA arrivals or departures lounges, and the arrivals lounges are landside at both T1 and T4.

Frankly, other than the Hilton, I would have thought that trying to get to any other airport hotel to use the gym and then getting back again in time to get to your flight is likely to prove to be more hassle than the benefit you'll get from it. But, as fbg says, much depends on how keen you are to get to a gym.

Immigration will give you no hassle, especially if you have your onward boarding pass - unless you're of a nationality that requires a transit visa to do this and you don't have one.
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