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Not all materials are able to reflect waves of a certain frequency. Usually this is limited to conductive materials which can react to the wave as if the material was a transmitting antenna efficient for that frequency (technically it must be an electric dipole). If your UFO is made of plastic with organic creatures inside, it will be transparent for waves and will reflect nothing.

Military are highly interested in uncooperative targets, ATC is not. It is possible ATC operators choose to not display PSR targets, only SSR targets. A reason is collision avoidance is based on altitude, and a target without an altitude cannot be managed, hence aircraft without a transponder able to indicate the aircraft altitude are usual not allowed to enter crowded airspace controlled by ATC. There are ATC operators on this stack, they may provide additional details about that.

Not all materials are able to reflect waves of a certain frequency. Usually this is limited to conductive materials which can react to the wave as if the material was a transmitting antenna efficient for that frequency (technically it must be an electric dipole). If your UFO is made of plastic or stealth material, it will be transparent for waves and will reflect nothing significant.

Military are highly interested in uncooperative targets, ATC is not. It is possible ATC operators choose to not display PSR targets, only SSR targets. A reason is collision avoidance is based on altitude, and a target without an altitude cannot be managed, hence aircraft without a transponder able to indicate the aircraft altitude are usually not allowed to enter a crowded airspace controlled by ATC. There are ATC operators on this stack, they may provide additional details about that.

However military have to detect uncooperative targets, and they use primary surveillance radars (PSR), the good old echo radar working by reflection on the target. In that case the radar station just sends a radio pulse and wait for any echo.

PSR stations use the Doppler shift to assess the target velocity. Stationary targets have no Doppler shift, the echo wave has the same frequency as the wave sent by the station. PSR stations may discard echos from stationary targets at low elevation, since most of them are just echos from buildings, trees, lakes, ground, etc. So your stationary object might be discarded and not visible.

Not all materials are able to reflect waves of a certain frequency. Usually this is limited to conductive materials which can react to the wave as if it was a transmitting antenna efficient for that frequency (technically it must be an electrical dipole). If your UFO is made of plastic with organic creatures inside, it will be permeable to waves and will reflect nothing.

In some countries, PSR and SSR stations are part of a network with a hub where all data from all stations are centralized, processed, correlated and redistributed to all ATC and military centers. In that case ATC operators will see an unidentified target detected by a PSR, without the exact altitude information (a PSR system is usually able to detect the direction, the velocity and the range).

Military are highly interested in uncooperative targets, ATC is not. It is possible ATC operators choose to not display PSR targets, only SSR targets. A reason is collision avoidance is based on altitude, and a target without an altitude cannot be managed. There are ATC operators on this stack, they may provide additional details about that.

ATC operator may not see the unidentified target. If they see it, the size won't be indicated, a target being basically a line segment (track) with SSR data (transponder id, altitude, speed, vertical speed) displayed as a label close to the track. As the operator who assigns the transponder id ("squawk code" in mode A/C) can associate this id with the aircraft flight plan and the flight number, often this flight number replaces the transponder id on ATC displays.

Source.

Whatever, if military detect an unidentified target, they will inform related civil ATC units. Actually it's possible a military controller will be present in the civil ATC unit, working with a civil controller to coordinate military and civil traffics.

However military have to detect uncooperative targets, and they use primary surveillance radars (PSR), the good old echo radar working by reflection on the target. In that case the radar station just sends a radio pulse and listen for any echo.

PSR stations use the Doppler shift to assess the target velocity. Targets stationary relatively to the radar station have no Doppler shift, the echo wave has the same frequency as the wave sent by the station. PSR stations may discard echos from stationary targets at low elevation, since most of them are just echos from buildings, trees, lakes, ground, towers, etc. So your stationary UFO might be discarded and not visible.

Not all materials are able to reflect waves of a certain frequency. Usually this is limited to conductive materials which can react to the wave as if the material was a transmitting antenna efficient for that frequency (technically it must be an electric dipole). If your UFO is made of plastic with organic creatures inside, it will be transparent for waves and will reflect nothing.

In some countries (notoriously the US, as stated in other answers and comments), PSR and SSR stations are part of a network with a hub where all data from all stations are centralized, processed, correlated and redistributed to all ATC and military centers. In that case ATC operators will see an unidentified target detected by a PSR, without the exact altitude information (a PSR system is usually able to detect the direction, the velocity and the range).

Military are highly interested in uncooperative targets, ATC is not. It is possible ATC operators choose to not display PSR targets, only SSR targets. A reason is collision avoidance is based on altitude, and a target without an altitude cannot be managed, hence aircraft without a transponder able to indicate the aircraft altitude are usual not allowed to enter crowded airspace controlled by ATC. There are ATC operators on this stack, they may provide additional details about that.

ATC operator may not see the unidentified target. If they see it, the size won't be indicated, a target being basically a line segment (track) with SSR data (transponder id, altitude, speed, vertical speed) displayed as a label close to the track. Stationary targets may be indicated using a specific symbol. As the operator who assigns the transponder id ("squawk code" in mode A/C) can associate this id with the aircraft flight plan and the flight number, often this flight number replaces the transponder id on ATC displays.

STARS ATC symbol for an aircraft, source.

Whatever, if military detect an unidentified target, they will inform related civil ATC units. Actually it's possible a military controller will be present in the civil ATC unit, working with a civil controller to coordinate military and civil traffics.

Questionable targets in the US are managed according to §5.3.2 of FAA Order JO 7110.65AA.

Many civil aircraft have a weather radar onboard, they may detect the uncooperative target, as these radars work by reflection on water droplets (rain in the cloud and below them). These radars show somehow the actual horizontal shape of the target by reflecting on droplets at any distance. However a solid target may be impermeable to waves, echos will be limited to the envelope portion facing the aircraft, the target will appear as a silhouette.

Many civil aircraft have a weather radar onboard, they may detect the uncooperative target, as these radars work by reflection on water droplets (rain in the cloud and below them). 

These radars show somehow the actual horizontal shape of the target by reflecting on droplets at any distance. However a solid target may be impermeable to waves, echos will be limited to the envelope portion facing the aircraft, the target will appear as a silhouette.

Weather radar pulses are sent close to the horizon, they don't illuminate targets in the ground direction. The weather radar of an aircraft flying at a high altitude won't see a UFO hovering close to the ground .