Unless the local cops object to your having such a license, the BATFE checks you out, if all is well, you get your license in 30 days or so.
It's important to point out that local law enforcement only has to be notified that you've applied for a C&R license- you do not have to wait for them to approve the license. It's not comparable to the approval that's required when purchasing certain NFA firearms.
This is correct. Per the BATFE Licensing Division, you are only notifying the local CLEO, not asking for permission or approval.
Current wait times are running about 2 months, and there are plenty of stupid things the BATFE is doing to slow down the process (my favorite: you mail your application to the payment center
in GA, then they mail
the current batch of applications to the data processing center in VA once a week
, where it takes a month
to scan the document into the system
before a human even gets to look at it). But overall, they are pretty much just gov't workers: not enthusiastic, but generally not interested in obstructing the process any.