How soon after a car payment is overdue can a bank repossess?

A simple answer to this question would be at almost any time.
Quite a bit will depend on the loan wording and agreement; if there is a grace period written in then it will fall under the grace period (often this is 30 day increments [30, 60 or 90]). However if there is reason to believe that their money or collateral is in risk they may have other reasons for repossession (i.e. A DUI, going into default, strange account activity, etc.).
In some cases you can go into default and it can be grounds for immediate repossession (though it may not happen immediately). Unfortunately you can, in many cases, go into default from missing or being late on a mortgage payment or credit card payment and they could repossess without ever being late on the car payment at all. In fact some shady dealers will wait for several more payments to come in, after the late one, before they repossess. This is not a very honest way to do business but it can happen.
Most likely they will let you be late in order to raise interest rates or add on extra fees or both before they attempt repossession and as long as you have kept good records of your payments and not been late often you have a decent chance of paying “In Good Faith” and they will not attempt a repossession as ultimately it could mean less money for them. (They’d rather have you pay late and pay more than to not pay at all.)
So to recap…If you have been late and there is no grace period repossession can take place at any time, and if there is a default clause you may not even need to be late to have repossession occur.
Read your loan policy carefully and look for a “Grace period” and for a “Default” clause to get an exact answer to your particular situation.
GOOD-LUCK!!!

that depends on the contract that the debtor signed with the creditor for the auto loan. typically--based on my experience--the creditor will send out the repossession order after two monthly payments have been missed.

2 months