Hey Eugene,
In NH, the way Tax Liens are handled will vary from town to town. I purchased a vacant lot in a NH town at a tax auction in 2019.
In that town the clock starts when they send out the first tax bill for the current year (in this case May). If the owner has not paid taxes by the following years tax cycle (again May in this case), then they give the owner 2 years and a day to pay the back taxes (all the while assessing penalties and fees). If after this period is up, there has been no attempt to pay, and no communication, then the town will simply take the property.
In that particular town, they usually wait till they have at least a couple of properties and then do a public auction (once a year, always in September).
My understanding is that the former owner has 3 years from the date the Town records the quit-claim deed to come back to the town and reclaim the property (by paying all back taxes, fees, penalties etc). After that 3 year period the former owner looses all rights to the property. I'm not sure what happens if the property was sold at auction before that 3 year period ends, and the owner comes back to take the property?
It's also worth noting that there is a 10 year "quite" period from the date the quit-claim deed is recorded, in which you cannot re-sell the property with a warranty deed (but you can resell with another quit-claim deed).
This particular town also does not sell tax liens to the public. They try their best to work with the owner, but if that fails, then they only ever take ownership of the property to sell at public auction.
In your case, I would reach out to the Town Administrator and ask them how tax liens are handled in that town.