Regarding the Heathrow “departure fees”, there are two parts to that, and both are rolled into the price of the ticket, not collected at the airport.
The big one for international travelers is the Air Passenger Duty (APD), which is high for US-bound flights because it is based on flight distance and the cabin flown (so it is higher for business class travelers). But—-and this is the important part—-it only applies to flights which originate within the UK. So if you are starting at an airport outside the UK (such as Lisbon) and merely connecting at Heathrow, the APD does NOT apply (as long as your Heathrow layover is under 24 hours).
https://onemileatatime.com/guides/uk-air-passenger-duty/
The smaller fee, which applies to all passengers, is the Passenger Charge, which supposedly goes to help cover airport services. Currently it is in the range of £32-£43, depending whether the most recent price increase has been approved.
https://onemileatatime.com/news/heathrow-raising-passenger-charge/
So, while one might have other reasons to avoid a connection through Heathrow—-it can be a confusing airport, especially if one must change terminals—- the Airport Passenger Duty is not a problem.
Edit: I just checked the price breakdown on British Airways tickets for 2 separate 2023 European trips.
Trip A has a 3-night stay in London on our way home, so APD’s do apply. Total taxes and fees on the ticket $352, of which $242.40 is the APD per person. (This is a Business Class flight, so APD is nearly double what it would be in Economy).
Trip B has our homebound flight originating at Bologna, with a 19-hour (overnight) layover at Heathrow. Total taxes and fees $103 per person.
This has been an important lesson for me, as I just learned we can save around $500 on our trips by planning our “London time” on our way TO Europe, rather than saving it for the end, on our way home.