score:6
Taxis that you can get from a taxi rank at the terminal (without booking ahead) are 'black cabs'. A black cab is probably big enough for 2 people plus 4 big bags, but might be a bit of a squeeze (depending on your definition of 'big'). Expect to pay £45 to £85 (depending on traffic) as per https://tfl.gov.uk/modes/taxis-and-minicabs/taxi-fares
You can get a good view of the size of a standard black cab in this video: http://youtu.be/YzvejjipQy0
Taxis that you can only book ahead of time are 'minicabs', there's a price comparator here: https://www.minicabit.com/search_trip.php Expect to pay about £40 - tell them about the luggage in advance so they don't send a vehicle that's too small. If you want a large, reputable minicab firm then you can do worse than Addison Lee: http://www.addisonlee.com/passenger-services/airport-transfers/heathrow-airport-transfers/
Uber exists in London and will charge you £53 for an UberXL vehicle from Heathrow to central London as per https://www.uber.com/cities/london
Personally, I'd get the Heathrow Express train to London Paddington and get a taxi from Paddington, it'll be cheaper (if you book tickets far enough ahead for the HE) and also quicker than a taxi all the way from Heathrow. (Traffic coming from Heathrow to Central London can be pretty terrible). It's pretty easy to get luggage on and off the HE, it's a large modern train designed to serve the airport. Step-free access, plenty of luggage space, enough time to load luggage, etc. https://www.heathrowexpress.com/tickets-deals/prices-fares Heathrow Express ticket prices start at £6.99 per person (if you book 3 months ahead), a taxi from Paddington to St Pancras should take about 10-15 mins and cost 10-15 pounds. HE journey time Heathrow->Paddington is 15 mins, trains leave every 15 mins. https://www.heathrowexpress.com/timetable-schedule/heathrow-london If you're coming from terminal 4 then you have to change trains, otherwise you don't.
You are absolutely right not to take 4 large suitcases on the Tube (London Underground), but the Heathrow Express is a different kettle of fish entirely.
Enjoy your trip!
Upvote:2
In your shoes I would get a pre-booked taxi service. You'll be able to put one or two bags on the back seat if they don't fit in the boot. If there is more than one of you travelling, you should say that you have a lot of luggage and a bigger vehicle will be provided.