It is strongly implied, but not definitively proven.
00:35:07 Mother numbered
00:35:09 So we'd always put them back in the right order.
This is typical OCD behavior. She had a preferred order, and the boys didn't respect that order, so she made it clear using numbering.
But, more importantly for your question:
00:35:11 Of course she did.
This established that this kind of behavior is to be expected of their mother, i.e. that she had many similar behaviors. This strongly suggests that she had OCD tendencies, but it's not enough for an ironclad diagnosis.
or was Adrian's and Ambrose's need for order such that she numbered the mugs so that her two sons wouldn't have to agonize about which mug went where?
Firstly, this doesn't make logical sense. If she numbered the cups, that means that without the numbers, the boys couldn't tell the cups apart. Because if they could do it without the numbering, the numbering would not have been necessary.
If they couldn't tell the cups apart without being numbered, they couldn't be agonizing over the "wrong" cup order to begin with.
Back when the cups weren't numbered yet, the alleged cause of the numbering (i.e. the boys being the ones who were upset about the wrong cup order) couldn't possibly exist, since the transcript makes it clear that the numbering was done to make the boys able to put the cups in the right order, which implies that they weren't able to do it before numbering the cups.
Secondly, that's not what the transcript tells us:
00:35:07 Mother numbered
00:35:09 So we'd always put them back in the right order
This states the intent of the numbering, rather than the reason for it. The ability to put the cups in the right order is the effect, not the cause, of the numbering.