Having been in a similar situation, both as an ATCO in the tower and a low hours PPL, , the correct instruction should have been "go around", pilot replies and complies, landing clearance to the following traffic PLUS traffic information as to what is happening ahead as it was obviously tight.
Seems pretty obvious the ATCO had misjudged the situation and had a bit of a moment, as acknowledged later.
The pilot, whilst having to comply with the go around instruction can always make a request as to how best get back to finals, otherwise they are quite entitled to fly the standard circuit.
As for the student scenario, any instructor that let's a student into the circuit without having practiced late go arounds should be shot.
I had the pleasure of practicing that in a very old Cessna 150 from a 40 degrees flap approach, heart rate of the first one went sky high, everyone should try it for a laugh.