I think it would work a lot better if nephew got his own card. You can send monies to Monzo via direct bank transfer, quoting correct references - straight to his account, check this out.
OR you can Monzo him money from your Monzo, which is instant as well. Seems like a lot less hassle than this log in, log out business you proposed! 
In case of total mess, if you’d have emergency access to his email account - only then you could log in to his Monzo on your own phone. I don’t recommend handing over password to email in general, but it could be done with lastpass and ‘emergency access’ functionality (you request password, wait set amount of time, if owner of password doesn’t deny access, you get password).