Citymapper , also available as an app for iOS and Android, has never failed me in London or other large cities where it is available.
Cabs, Uber, Tube, buses, Heathrow Express, cycling, walking, I don't know of any mode of transport they don't cover, at least in London.
The mobile app (but not the website) even finds a combination of public transit and cars-to-hire (scroll to the bottom of possibilities offered):
It also shows you the cost and the time of any alternative routes it finds, the best place to be on trains, as well as the best entrances and exits for tube stations.
And it calculates time for transfers inside the stations.
It's my most important app whenever I'm there.
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The simple answer is that depending on the address Heathrow Express or a taxi may not be the fastest way to get to your address and routing apps are taking that into account.
London is highly congested, road speeds are often 8mph for cars, so taxis can get stuck in that traffic.
Heathrow Express goes into Paddington Station on the western side of central London, which is fine if you have accommodations on that side of the town.
However if your destination is elsewhere by the time you have got off at Paddington and then found the correct route to continue your onward journey
the time saved by using the Express can be eaten up.
If you can give us some idea of where the address is further it may be possible to give more focused help.
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On Google Maps the Heathrow Express will show up if you specify a terminal to depart from.
As for Taxi journeys, time wise those would mostly be equivalent to the Car timings on google maps.
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As comments mention Rome2rio is a surprisingly good solution within a city as well.
I never considered it for anything but intercity travel.
Google does find the Heathrow Express
but you need to kind of force it: if you just enter Heathrow Airport it won't find it
but if you enter say Heathrow Terminal 5
then it will.
It also suffers from the problem of trying to find the best solution at the minute and a cross
London trip has so many solutions that a better one five minutes later will be crowded out.
Peruse the schedule explorer option.
Also don't forget Google Maps doesn't add time for underground transfers which can be long (or not) -- this adds a lot of manual work to the search.
These 3D maps of Underground stations can help.
As for using a cab, it's very likely a car can't beat the underground so getting as close to your destination as you can with it
is a good idea and then swapping to a cab might be good but you need to manually search the station in the vicinity of your destination.
Search for your destination first in Google Maps, then click the [x] clear search button and enter underground station to find the underground stations in the vicinity.
Also note the difference between the traditional Hackney carriages and Private hire (minicabs) .
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Transport for London's journey planner at https://tfl.gov.uk/ takes all public transport in London into account, so includes the Heathrow Express (as well as more outre options such as the river bus and the cross-Thames cable car).
It doesn't, however, include cabs, probably because these are very expensive and of highly-varied travel times.
The latter complaint could also be made of buses, but TFL's are all fitted with location trackers so TFL has a good view into congestion insofar as it affects the buses.
This doesn't solve the entire problem as specified, though;
sorry.
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Neither Google Maps nor Traveline.info seems to consider neither Heathrow Express nor cabs. Is there any service that would?
How do I find the fastest route from Heathrow to an address in London using all forms of transport?
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Transport for London's journey planner at  https://tfl.gov.uk  takes all public transport in London into account, so includes the Heathrow Express (as well as other options such as the river bus and the cross-Thames cable car).  Rome2rio and Citymapper are also recommended. Heathrow Express is shown on Google Maps if a departure terminal is selected. Google shows the Heathrow Express if you specify a terminal and you can see car journey lengths for taxis. Citymapper, TfL and Rome2rio are alternatives for journey planning.
Citymapper, TfL journey planner and Rome2rio are all solutions. Google finds the Heathrow Express if you specify a terminal and taxi journeys would be equivalent to cars.