For me the responsibility of cleaning a company vehicle both on the inside and outside lies with the employee,
Especially if they have sole use of the work vehicle
If not then the company needs to make sure the last person using the vehicle leaves it in a clean condition for the next user
With regards to who's time is used, well I'd say if it's a commercial van that is also used for commuting it should be cleaned on company time
but if it's a car that's used mainly for commuting and personal use then it's cleaned on the employees time
As far as who pays well it depends, if there is space on the premises, then provide the employees with the equipment to clean
If not then the company pays for the use of such equipment at a garage forecourt
That's just my opinion and how we ran our fleet of 20+ vehicles back in the day, others might differ in their practices