Some context:
when things finish (until, by the time..., etc.):
Fotini wrote and phoned for three months, at which point she lost patience.
She was determined to fight on until she got some satisfaction. (this is ok)
She finally contacted a solicitor, by which time the hotel had closed down.
�at which point� means �then� at that moment, at that point in time.
�by which time� means �by then� by that time
Question 1: What�s the difference between these expressions?
The difference is �at which point� is defining a specific (point in) time.
Whereas �by which time� is giving the end of a period of elapsed time.
Question 2: What verb tenses can I use with both expressions.
For at which point they used simple past, matching the simple past
actions that preceded the losing patience (like a list of actions:
she wrote, phoned, and lost patience). You cannot use had lost patience
because the losing patience just happened. Had lost patience
would mean that the patience had been lost earlier, but it was not
lost earlier. You must use had closed down with by which time
because the closing down happened before the solicitor was contacted.