How close in time or distance will ATC let a pilot fly to the localizer, before cancelling the approach clearance, where the pilot has not reported being established? Is it the controller's call when to cancel the clearance?
Each instrument approach consists of a lateral part and a vertical part, in the case of the ILS approach these are: localizer and glideslope.
Aircraft are usually vectored to intercept the localizer approximately 1-2nm before the Final Approach Fix (FAF), where the glideslope is captured. Aircraft can be vectored onto the localizer as far as 20-25nm out, depending on terrain, localizer equipment and many other factors. If an aircraft is not established on the localizer, there is no point in continuing the approach, so it will be cancelled and the aircraft revectored.
If the aircraft is established on the localizer, it will try to intercept the glideslope, which is the vertical guidance system in the ILS approach. For VOR/DME, LOC/DME, NDB/DME or RNAV approaches, the vertical guidance is either achieved through correlation of location to a specific altitude in the chart or technical means which do not require a glideslope.
If an aircraft passes the Final Approach Fix and does not seem to descend, it will be considered as not established and the approach clearance will be cancelled, instructing the aircraft to either follow the missed approach procedure or manually vectoring the aircraft for another approach.
I'm not 100% sure that I understand your question, but I think you're asking this: if a pilot is on a localizer or ILS approach but doesn't report established, when will ATC cancel his clearance and instruct him to go missed or vector him? But you didn't clarify if the pilot is actually flying the approach correctly or not, where he started the approach, if he's being vectored, what his current clearance is etc. so I'm not sure I understand the scenario.
Anyway, I couldn't find a definitive, objective answer to this and I'd be surprised if there is one. If a pilot deviates from an approach procedure in any way, then the controller has to make a judgement call based on current traffic, workload, terrain, student pilots etc. and I think it's unlikely that there could be a single, definite rule on what to do when.
Apart from any other considerations, if the pilot doesn't report established it could be a comms failure and there are procedures for that: in this case since he was already cleared for the approach (presumably) the controller will expect him to continue. And if the pilot deviates without comms, there isn't much the controller can do anyway apart from clear the surrounding airspace.
Having said that, the FAA procedures for controllers give this instruction for handling aircraft on a radar arrival - i.e. under direct ATC control - and it seems plausible that they would apply it in other cases too:
b. If deviations from the final approach course are observed after initial course interception, apply the following:
Outside the approach gate: apply procedures in accordance with subpara a, if necessary, vector the aircraft for another approach.
Inside the approach gate: inform the pilot of the aircraft's position and ask intentions.
"subpara a" simply says that the controller should provide suitable vectors with a maximum 30° intercept angle to the final approach course. The approach gate is defined in the P/CG:
APPROACH GATE- An imaginary point used within ATC as a basis for vectoring aircraft to the final approach course. The gate will be established along the final approach course 1 mile from the final approach fix on the side away from the airport and will be no closer than 5 miles from the landing threshold.
The question doesn't make sense, because the clearance won't be issued until the aircraft is established on a published segment of the approach. Nevertheless, I will try to explain how it works.
Note that the aircraft can only be given vectors to final if its on radar, and so the controller can see the aircraft's exact position. The controller can't, however, tell if the localizer needle is alive and the frequency identified, so he asks "Report established."
It is only AFTER the aircraft is established on a published segment of the approach that the controller can say "You're 5 miles from MOXIE, cleared ILS 27 approach."
If the aircraft is outside the approach gate and can intercept another vector to final without requiring too sharp a turn, the controller will continue to provide vectors to final. If the aircraft cannot be vectored from where it is, the controller will spin the aircraft back downwind or tell him to fly the missed, and give the pilot another shot at getting established.
If the aircraft is inside the approach gate, but deviating from the final approach course/localizer, the controller will say something like "3 miles from the airport 1 mile right of course, say intentions," and the pilot can ask for vectors downwind and back around onto final. He won't have his approach clearance terminated.