I presume you are referring to airfields which do NOT have a licensed ATSU, ie A/G or AFISO.
So does an aircraft doing circuits that has only just climbed out from take off count as an aircraft intending to land at that aerodrome, and so who has right of way etc.
Yes - Rule 16 (I think I have the right number!) applies - conform to traffic pattern etc.
Similarly, if an another aircraft was to try to join base leg, whilst the student or any pilot for that matter was downwind/late downwind, what would the rights of way be?
Same applies - the base leg joiner should not, in my opinion, be joining on base leg! (This has been discussed comprehensively on another thread!)
Also rules say that if two aircraft are approaching to land the lower aircraft has priority (emergencies excluded) except you should not cut in front of an aircraft which is already established on approach.
From a practical point of view instructors should teach "Go-Arounds" from Downwind, Base Leg AND Final which in certain circumstances you may have to do due other traffic.
Non Standard Cicuit Joins