A motor vehicle is a chattel and you do not have to wait until a grant of probate or letters of administration have been issued to be able to transfer a car to another owner or to sell it. You will need to deal with the car tax as this can no longer be transferred to another person, it must be cancelled and re set in the name of the new registered keeper. Further information can be found on the gov.uk website
There is information on this link to the government website about the forms and documents that are needed if you either give the car to someone else or sell it click here.
If a car has to be moved before ownership is transferred, the person who drives the car must have insurance that enables them to do so. Usually the driver's insurance will be invalid unless you have made contact with the insurance company to arrange cover.
If the car is old and you feel it is not worth trying to sell it commercially, you may want to consider using a scheme that will scrap and recycle the car to raise money for charity. You will be given an official DVLA destruction certificate to prove that it has been dealt with by a recycling facility that has been certified by the Environment Agency.
Charity Car is operated by a commercial company called CarTakeBack.com which enables the car to be scrapped for recycling and they commit to pass all of the value to one of their partner charities; you can choose which charity from the list given on their website. Charity Car is not itself a charity. If your preferred charity is not listed on the website you can ask them to register using a form on the website
Giveacar is a not for profit organisation which will either sell a car at auction or scrap it and raise money for your chosen charity. If the charity you would like to donate to is not listed on their website, you can ask the charity to register via the website.