I think you have the right to ask to go part-time, but your Employer does not have to grant your request.There may be exceptions to this for example, if you look after young children, and your request was denied, and you found out some one else is working part time for the same reason then you could have a claim for discrimination. But in reality this would probably lead to trouble in the workplace. If your Employer granted your request to go part-time there is no guarantee you could revert back to full-time work, if your Employer didn't want you to.
All part time workers have the same rights as full-time workers, regarding pay, holidays etc, and obviously on a pro rat basis.
There maybe some legislation put in place by the Govt because of the Corona Virus outbreak, that will protect the rights of workers that may have to look after their children because of the Govt action to close schools, we will have to see. You could contact your MP and ask him to put pressure on the Govt, as I am sure he/she must be aware of the situation.
Incidentally Job sharing is a different matter altogether, even though it does allow employee's to , on the face of it work on a part time basis, the Employer's interests are covered on a full time basis.