You could get a man and van hire company to drive you there and help unload and get the luggage into the airport. To them it won't make much difference whether they're carrying moving boxes and furniture or suitcases. In fact, suitcases are easier to carry. If there's two of you, you can use two trolleys easily.
Most of these companies charge per the hour, anywhere from around 20 to 50 GBP per hour. They are usually meant for moving and removals, but I'm sure they'll take an easy job such as this. You can offer to pay for the parking too.
I've googled "Luton Man and Van" and went on the first non-add link here. This is 30 GBP per hour for one man and one van, two hours minimum.
Or you could get a large taxi/uber and offer the driver 10 or 20 GBP cash to help you carry all the stuff inside.