Many thanks to Prospero for this really thorough and fascinating analysis, and for all of the hard work that went into it.

As it happens, a couple of weeks back it occurred to me to do something vaguely similar, although I only had limited time and so used a short-cut: Wikipedia's lists of airlines and destinations are usually a pretty reliable census of direct flights. (It's worth remembering that "direct" is not necessarily the same as "non-stop".) But it may contain errors, so it isn't as reliable a method as a real-time interrogation of current schedule data from a GDS. Nevertheless, it was probably good enough to get a broad picture.

Another short-cut that I took was basically only to count routes. That means that this analysis was not as sophisticated as Prospero's, which has the additional information about frequencies and therefore gives a better guide to the extent of BA's dominance.

I took every destination listed in Wikipedia, even though some are yet to start, some are due to end, and some are seasonal: these are features of BA's route network as much as other airlines', so much of it would come out in the wash anyway.

I took cities as destinations, including counting JFK and EWR as "New York".

So, first: the data.

For LHR, there are 307 airline/destination combinations (for example, there are 5 airlines flying direct from LHR to DXB, so that's 5 combinations) across 198 different destination cities.

116 of those combinations are destinations served by only one airline.

48 of those "monopoly" destinations (24% of total LHR destinations) are served direct by a non-BA airline. These are A Coruńa (Vueling), Addis Ababa (Ethiopian Airlines), Ahmedabad (Air India), Algiers (Air Algérie), Ashgabat (Turkmenistan Airlines), Astana (Air Astana), Asturias (Iberia Express), Auckland (Air New Zealand), Baku (Azerbaijan Airlines), Bandar Seri Begawan (Royal Brunei Airlines), Barbados (Virgin Atlantic), Belgrade (Air Serbia), Bogotá (Avianca), Casablanca (Royal Air Maroc), Charlotte (American Airlines), Cologne/Bonn (Eurowings), Colombo (SriLankan Airlines), Cork (Aer Lingus), Detroit (Delta Air Lines), Dhaka (Biman Bangladesh), Guangzhou (China Southern Airlines), Halifax (Air Canada), Hanoi (Vietnam Airlines), Ho Chi Minh City (Vietnam Airlines), Islamabad (Pakistan International), Jakarta (Garuda Indonesia), Karachi (Pakistan International), Lahore (Pakistan International), Luxor (EgyptAir), Malta (Air Malta), Manila (Philippine Airlines), Marrakech (Royal Air Maroc), Mauritius (Air Mauritius), Medina (Saudia), Melbourne (Qantas), Minneapolis/St Paul (Delta Air Lines), Ottawa (Air Canada), Perth (Qantas), Portland (OR), (Delta Air Lines), Rabat (Royal Air Maroc), Raleigh/Durham (American Airlines), Rijeka (Croatia Airlines), Salt Lake City (Delta Air Lines), Shannon (Aer Lingus), St John's (Air Canada), Taipei (EVA Air), Tashkent (Uzbekistan Airways), Tunis (Tunisair).

There are 68 "monopoly" destinations (34% of total LHR destinations) served direct only by BA. These are Abuja, Accra, Austin, Baltimore, Bangalore, Barcelona, Basel, Bergen, Biarritz, Bilbao, Billund, Bologna, Brindisi, Budapest, Buenos Aires, Cape Town, Chania, Chennai, Corfu, Denver, Faro, Gibraltar, Glasgow, Gothenburg, Grand Cayman, Hanover, Hyderabad, Ibiza, Innsbruck, Inverness, Kalamata, Kiev, Kos, Kraków, Las Vegas, Leeds/Bradford, Luanda, Luxembourg, Lyon, Málaga, Manchester, Marseille, Menorca, Montpellier, Murcia, Mykonos, Nantes, Nassau, New Orleans, Newcastle upon Tyne, Nice, Olbia, Palermo, Phoenix, Pisa, Prague, Pula, Rio de Janeiro, Salzburg, San Diego, San Jose (CA), Santiago de Chile, Santorini, St Petersburg, Tallinn, Toulouse, Venice, Zakynthos.

However, 47 of these destinations are in any event served direct from other London airports. So the list of destinations to which BA has a "monopoly" from LHR, and to which there is no alternative direct service from London, is only this shorter list of 21 (11% of LHR destinations): Abuja, Accra, Austin, Baltimore, Bangalore, Buenos Aires, Chennai, Grand Cayman, Hyderabad, Leeds/Bradford, Luanda, Manchester, Nassau, New Orleans, Newcastle upon Tyne, Phoenix, Rio de Janeiro, San Diego, San Jose (CA), Santiago de Chile, St Petersburg.

FWIW, for the 47 destinations which have non-BA direct competition from other London airports, these are the number of competing non-BA airlines from the other airports, so in many cases there must be reasonable competition amongst the non-BA airlines as well as between them and BA:-
Barcelona: LGW (4), LTN (2), STN (1)
Basel: LGW (1), LTN (1), STN (1), LCY (1)
Bergen: LGW (1)
Biarritz: LGW (1), LTN (1), STN (1)
Bilbao: LGW (1), STN (1)
Billund: STN (1)
Bologna: LGW (1), STN (1)
Brindisi: LGW (1), STN (1)
Budapest: LGW (1), LTN (1), STN (1)
Cape Town: LGW (1)
Chania: LGW (4), STN (1)
Corfu: LGW (5), LTN (3), STN (4)
Denver: LGW (1)
Faro: LGW (5), LTN (4), STN (3)
Gibraltar: LGW (2), LTN (1)
Glasgow: LGW (1), LTN (1), STN (2)
Gothenburg: LGW (1), STN (1)
Hanover: STN (1)
Ibiza: LGW (5), LTN (2), STN (5)
Innsbruck: LGW (5), LTN (1), STN (2)
Inverness: LGW (1), STN (1)
Kalamata: LGW (2)
Kiev: LGW (1), LTN (1), STN (1)
Kos: LGW (4), STN (3)
Kraków: LGW (1), STN (1)
Las Vegas: LGW (2), STN (1)
Luxembourg: STN (1), LCY (1)
Lyon: LGW (3)
Málaga: LGW (4), LTN (3), STN (3)
Marseille: LGW (1), LTN (1), STN (1)
Menorca: LGW (1), LTN (3), STN (4)
Montpellier: LGW (1), LTN (1)
Murcia: LGW (1), LTN (1), STN (1)
Mykonos: LGW (2), LTN (1)
Nantes: LGW (1), LTN (1)
Nice: LGW (2), LTN (1), STN (2)
Olbia: LGW (4), LTN (1)
Palermo: LGW (1), STN (1)
Pisa: LGW (1), LTN (1), STN (1)
Prague: LGW (2), LTN (1), STN (2)
Pula: LGW (2), STN (2)
Salzburg: LGW (5), LTN (2), STN (4)
Santorini: LGW (5)
Tallinn: LGW (1), STN (1)
Toulouse: LGW (2), LTN (1), STN (1)
Venice: LGW (3), LTN (1)
Zakynthos: LGW (5), LTN (1), STN (4)