Nanny-state = over protective state that enact laws to protect its citizen even if said citizen did not ask for protection
Puritanical = strict laws based on a set of morals, often religious in nature.
Nanny state has nothing to do with morals. Nanny state enact laws to protect you even if you don't want it to. These laws are evidence based, not morality driven. Thus puritanical nanny state is not a possibility.
Your examples of puritanical nanny-statism have nothing puritanical.
First, where did you get your definition of Puritanical from? Puritanical is not a set of strict laws, as it can be used for non-law things. Puritans had strict laws, but when you use puritanical its an adjective to say strict.
Also, nanny-state has everything to do with morals. You seem to be under the impression that morals means something other than this "a person's standards of behavior or beliefs concerning what is and is not acceptable for them to do." So the nanny-state is a state that is concerned with the behavior and beliefs in terms of what is acceptable.
Lets see some examples of how the word can be used.
"My parents are extremely puritanical and will not allow me to date."
"Dancing didn’t fit into the puritanical ideals of morality."
See how its being used?
Now, check out this link and how it describes puritanical.
https://www.collinsdictionary.com/dictionary/english/puritanical
If you describe someone as puritanical, you mean that they have very strict moral principles, and often try to make other people behave in a more moral way.
Now, if a state had very strong moral principles they would want to enact strict laws to force people to behave morally. A nanny-state is a state that believes people should be forced to not be able to do unacceptable things. The only way you can define what is unacceptable and or acceptable is to have a set of morals. Thus a nanny-state going off of what it thinks is moral aka acceptable or unacceptable will have laws adhering to its morals.