Hi Bob,
Transporting queen cells is a bit tricky. Temperature control is important but not critical. What is critical is the state of the cell and handling. There is a period between days 11 and 14 where the pupating larvae inside is very delicate. She has a teeny tiny umbilical between the pupae and the jelly pot. Small disturbances can sever the cord and fail the cell. Also during this period the most delicate and most essential part of the queen, her wings, are developing. Handling can damage the wing buds and result in a queen that cannot fly properly or at all for mating. So, the ride needs to be day 14/15, warm, and cushioned. For the ride, put the cell in your breast pocket under a vest and drive her over like that. Your breast is warm and rides smoothly relative the rest of the car.
More information is needed to cover this properly, specifically how long the hive has been queen less and how big the hive is in population and space currently. Going on the assumption that it just went queen less within a few days ago..
It is only 15 minutes away and there is only one. if I may make an alternative suggestions to consider.
Option1: Have the someone bring you a nuc box, or the whole hive if it is a single, of bees to place in your bee yard. Do the cell transfer and installation a few steps from your source colony on day 14. Leave the nuc at your place, do not move it. On day 17 check that the queen has emerged properly in the nuc. Then the someone can take that nuc home and wait out for her to mate and start laying, another 8 to 14 days. Once laying, combine the laying nuc into the hive by newspaper combine. Towards the end of this timeline if the queen less hive was abandoned and sitting waiting, it will go LW. Either give it brood frames along the way or combine it with a queen rite hive to keep it stable.
Option2: Place the frame with the cell in your nuc box on day 9 (you have at least 1 right? cannot call yourself a beekeeper without having one around) , wait until your calendar says day 14. Then take the nuc over to the someone place, riding on the front seat of the car, put a cell protector over the queen cell, and install the whole frame into the receiving hive. check for emergence day 17. Wait 8 to 14 days after emergence to go look for eggs.
Option3: If the hive has just very recent gone queen less, it may successfully raise its own queen by just giving it a frame of open brood from your hive. (this is simplest and prevents LW condition from developing)
Option4: If the hive is queen less, depending on how long the condition existed, and if your queen cell is not already just a few days from being ripe - there may be a high probability that the hive will go laying worker and kill the cell before the hive is turned around. Abort the queen cell idea and instruct the someone to combine the queen less hive with a queen rite hive to stabilize it. Use a queen excluder and newspaper when combining to slow the bees merge down and also to trap the possibility of a rogue virgin already being in there. Then go back to it 7 to 10 days, do a normal split and install your cell in the split or install a mated queen (caged)
Need some dates - timeline from you to guide which option will give you the highest probability of success. How long has the hive been queen less, what is the stage of development of your cell (day#).
Btw: In my bee yard, unless I know exactly what is going on; a queen less condition is always taken care of by option4. Combine then split later. This method is pretty much guaranteed success, as it is using the bees to sort out the bees (LWs are killed) and keeps the colony(s) stable until a new queen or cell is ready.
Hope that helps.