taxis are not quite my thing, but there's
a report that london hackney carriage ('black cab') fares went up in 2022 and "fares to Heathrow Airport will rise as well." - I think that heathrow charge taxis a fee for picking up or dropping off there, so of course that gets added on. (hackney carriage fares are set by the local licensing authority)
hackney carriage fares are a mix of distance and time, so even if you're stuck in a traffic jam, the meter will tick on.
private hire vehicle (minicab) fares are not regulated at all, they can charge what they like and vary their fares depending on how busy it is. My understanding is that there are less minicabs / minicab drivers around than there were before the pandemic (there's a lot of lines of work short of people at the moment - most bus operators are struggling to get drivers at the moment) so that may have pushed fares up a bit. And again, if they get charged for going in to Heathrow, that will add to the price.
Without knowing just where you are, is it worth looking at bus and train services next time? (or were you travelling on a strike day?)
I don't think it's quite as comprehensive as it was before the plague, but there's quite a network of bus routes in to Heathrow both from inside London and Bucks / Berks / Surrey - trains from the west is a slightly different matter. And most of those buses run quite early and late in the day, and many of TFL's run all night (Slough - Heathrow did run all night before covid, not sure if it came back) - even if you're somewhere obscure, would taxi to / from somewhere like slough or staines or ealing or hounslow work?