When there is a holding pattern assigned by a civilian airports ATC, do military aircraft get first priority over landing?
Do military aircraft get landing priority?
  


If the government so wishes, yes. ICAO Doc 4444 says:


  7.6.3 Priority for landing
  
  7.6.3.1 If an aircraft enters an aerodrome traffic circuit without proper authorization, it shall be permitted to land if its actions indicate that it so desires. If circumstances warrant, aircraft which are in contact with the controller may be instructed by the controller to give way so as to remove as soon as possible the hazard introduced by such unauthorized operation. In no case shall permission to land be withheld indefinitely.
  
  7.6.3.2 In cases of emergency it may be necessary, in the interests of safety, for an aircraft to enter a traffic circuit and effect a landing without proper authorization. Controllers should recognize the possibilities of emergency action and render all assistance possible.
  
  7.6.3.3 Priority shall be given to:
  
  
    a) an aircraft which anticipates being compelled to land because of factors affecting the safe operation of the aircraft (engine failure, shortage of fuel, etc.);
    
    b) hospital aircraft or aircraft carrying any sick or seriously injured persons requiring urgent medical attention;
    
    c) aircraft engaged in search and rescue operations; and
    
    d) other aircraft as may be determined by the appropriate authority
  


Or c), if by military it's a search and rescue C-130 (for example) wanting to land to refuel and be on alert ASAP.

Now is it wise to put d) ahead of a)? Of course not. So if there is a prioritized government VIP, they will have to wait for the low-on-fuel emergency or the rogue-in-distress (first two paragraphs).

If it's a fighter, say an F-35, why would it land in Heathrow for example among civilian Jumbos? Unless of course it's been coordinated beforehand, or there is an emergency, then yes it will land first.

In the documentary Jetstream, RCAF F/A-18's flew from Canada to KLAX and landed there during heavy traffic, they were treated as normal planes when it came to the sequencing. It was part of their training to manage the complex aircraft in the heavy traffic environment of the civilian world.

You can watch it here. The narrator says (around 3:40):


  In Cold Lake an F-18 coming in to land is top priority, but here you are just another plane in a long queue.