We are going to be in York for a day's visit, and plan to check our luggage at the left luggage facility in the train station. Does anyone know how long lines might be on a Friday morning and for pickup in the late afternoon?

Depend on when you are going. I was just there in March and there was no wait for drop off or pick-up - their signs recommend 10 minutes, but I would imagine in the summer or during a holiday they could be much busier.

Of the dozens of times i've passed by the Left Luggage in York station there was nary anyone in sight - sometimes including an agent

Though it can just be your bad luck that on the day you turn up there happen to be several families with large amount of luggage in the line in front of you, who are arguing with the agent about the cost or damaged/missing items!

This is completetly unpredictable.

Most left luggage operations require bags to be xrayed, and use the same person both to accept/scan incoming bags and hand out bags being withdrawn.

It's not uncommon for the person in the queue ahead of you to be gobsmacked at the cost of having left two bags for 25 hours, or anything else that might have gone (or thought to have gone) wrong.

Hardly matters depositing bags: but personally I always aim to collect them 15 mins before departure (an age in most British railways stations, where there's never any need to arrive more than 90 secs before departure: only Waterloo and Victoria even need more than 45 secs to find the platform)