When adding a user to a group in AD to give them access to files on a storage server typically you need to reboot or log out/in to gain access to the share.
I need to test further but looks like these two commands will refresh your access without relogging in or rebooting:

klist purge
klist tgt

After that you can run the following command to check if the groups changes:

whoami /groups
gpresult /r

REFERENCE: http://woshub.com/how-to-refresh-ad-groups-membership-without-user-logoff/