The ATC is unable to provide radar service to pilots, what's the standard way to inform this?
Without checking the Controller's Handbook, what I hear most often are two things:
a. Standby and remain clear of class [b, c] airspace, for VFR operations.
b. (surveillance) radar services not available, squawk xxxx, report yyyy
c. Radar out of service, N1234, radar service terminated.
An example of b was when going to an airport near DCA, radar was out of service.  The controller handled my flight but relied on talking with a controller from Baltimore. This was a few years ago, and I now understand that the remote radar can be transmitted to other controllers taking a remote controller out of the loop.
The ATC Handbook is FAA JO 7110.65W, https://www.faa.gov/documentlibrary/media/order/atc.pdf
Radar can be limited due to equipment outages, and also the display available to the controller.  For example tower displays are for orientation and not certified for separation and other services.
The order states that controllers may advise that:
PRIMARY RADAR UNAVAILABLE (location)  or
RADAR SERVICES AVAILABLE ON TRANSPONDER EQUIPPED AIRCRAFT ONLY
Since there are many conditions for radar not being available and since an aircraft can be in various states (VFR/IFR, approach, visual approach, etc.) it would be best to consult the Order for the recommended phraseology.  It is noteworthy that CONUS radar coverage has improved since the Order, and as a result there is often backup coverage for many locations which could influence services which can be offered.
Finally, operationally ATC still has local and regional variations, which may influence the options and corresponding phraseology utilized.