Tried and tested companies include www.handicars.co.uk www.simplyairports.co.uk www.justairports.com www.london-transfers.com and www.exclusiveairports.co.uk It's not unusual for credit card payments to attract a small surcharge, so a cash payment will save you a few pounds.
I've used Just and Simply and have been satisfied with both.
Ive booked Dot2 Dot a door to door shuttle service for friends and guests and they have been very satisfied. They do early pick ups.
If there are 2 or more of you it might be cheaper to book a local minicab company.
If you're on a budget then depending where you are staying (and probably which terminal you're flying from) there is a night bus that goes to Heathrow too.
A black taxi to the airport that you waved down in the street would cost you about £50 to £60. Put your time of departure and addresses into this website calculator to get a more accurate estimate
As was said above the "N9" bus runs through the night and early mornings and costs £2 per passenger. They do have luggage space. But they would take about 80 minutes plus the time waiting for one to arrive.
Dot2dot is getting more and more complains. I always recommend Simply. They are cheap and efficient.
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