As Marco mentioned above, the issue is not "bank owned" vs "non-bank owned" ATMs, the issue is Dynamic Currency Conversion or DCC. This is where the entity operating the transaction point offers to do the currency conversion for you, as opposed to having the network or your credit card do it for you. Of course they charge a rate that will cost you from 3% to 10% vs your ATM card rate of about 1% plus bank fees (if any).
To be fair, yes, I have encountered this at Heathrow, but also at ATMs in London, At ATMs at the Barcelona Airport, ATMs in Barcelona and surrounding towns, At Tegel Airport in Berlin, and at ATMs at Berlin HBF. The ATMs were a mix of Travelex, Bank ATMs, and Private Label. You can also encounter this with a credit card at a merchant.
HOWEVER....In each of these cases, I walked away paying no more than my typical ~1% conversion fee, because they specifically ask you if you want to do the transaction in your home currency (bad) or the local currency (good), or more simply, decline the DCC. Yes, it may seem confusing, but you better get used to it, in one to two trips a year to Europe, I now encounter it nearly every trip.