No, NID wouldn't change the ASM paths.
I presume that you did set the ORACLE_SID to CRAMSIT on the non-RAC server before the restore.
What you should have done is to set DB_UNIQUE_NAME='CRAMSIT' (even with DB_NAME='CRAMPRD') on the non-RAC server before the restore. Oracle would have used the DB_UNIQUE_NAME to set the ASM PAth (note that during the restore it would have shown "restoring to +DATA1/CRAMPRD" but would have actually been restoring to "+DATA1/CRAMSIT").
You will now have to use Oracle Support Note
How to rename/move a datafile in the same ASM diskgroup [ID 564993.1]
to rename the paths in ASM.
Hemant K Chitale
You can use
set newname for datafile 3 to '+DATA_DG/.../undotbs1.dbf';
for example ...
Thanks Hemant for your quick response and comments.
Will follow the Note