I will be arriving in Heathrow around noon with a lot of luggage (around 4 big pieces). Will I find room in taxis to take this to downtown London or should I book something in advance ?
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!