Leaving from LHR you will get an email about 48 hours before the flight to confirm your booking. There is a choice of car there and a contact number which you can use if you want anything else. When arriving there is a small 'lounge' in arrivals (landslide) where you go to 'check in' for your CD. I presume you can ask there if the car they have allocated can take the 4 bags. My experience is that 90% of the time you get a Mercedes station wagon so it should be no problem anyway.

I have only used CD in Dubai linked with Dunbai Connect. I just went to the CD desk and they allocated a car. When leaving the hotel (Meridian) organised the car and as there were several people needing transport we got a people carrier sort of vehicle (which could take lots of luggage).