As
RussianTexan has seemingly already discovered, Barclays have a branch
and (standalone) cash machine locator on their website
here.
At Heathrow they have machines listed at "Terminal 5" (though no indication of airside, landside or both), and also at "Teminal 2 airside" and also "Travelex Departure Gate 2, Terminal 2". As has already been pointed out, the cash machines listed at T2 pose something of an existential dilemma, given that T2 has been demolished.
One option would be for the OP to get on a bus (for free) from T4 to Hatton Cross and visit the Barclays cash machines at the branch there, before then proceeding into central London by Tube - bus routes 482, 490, 555 and 557 will take you this short hop (see this
bus spider map), and Hatton Cross is fairly obvious once you get there (also there may be announcements on the bus, if it's one of the TfL routes, or otherwise you could ask for a heads up from the driver).
The Barclays branch at Hatton Cross is just over the road from the Tube and bus station - there's no Google Streetview within the airport estate but you can see the bank building just off the roundabout, in relation to Hatton Cross station, on this
Bing maps bird's eye view (the bank has the pin point stuck in it - you can also see a single-decker red bus in the bus station in front of the Tube station).
Alternatively you could get a bus the other way from T4 to T5, but it's three times as far, there's one less bus route that goes that way, there's a slightly less frequent Tube service from T5 and anyway you'll only end up going through Hatton Cross in the end.
Of course, if you're
not taking the Tube into central London (and are taking the HEx), that does changes matters!
All that said,
rwoman's suggestion of heading into London first and then finding a cash machine might be the better one. Barclays are one of the 'big four' high street banks here, so they have cash machines all over the place.