Alexey from tinyCam says it is caused by the Stream Profile setting.
The choices are AUTO, MAIN and SUB.
It should be set to MAIN.
AUTO uses MAIN (1080) for single camera views and SUB (360) for multi camera views.
I was viewing 4 cameras at once in a browser which would set all of them to 360p.
I have switched Stream Profile to MAIN and the cameras now stay set to HD.