Hi! In general, I recommend buying tickets/reserving time slots in advance for everything that you really want to visit while you're here. If it's something that you'd be disappointed if you left without visiting, book it!
FYI, Winter Break for schools in France is on rotating weeks between February 7th and March 9th, and 2 out of the 3 regions (including the Paris region) will be on break from February 22nd to March 9th, so wait times will likely be longer.
Another FYI, "Tours de Notre Dame", means "Towers of Notre Dame", so if you see "Tours de Notre Dame", it's referring to visiting the (bell) towers of Notre Dame, not a guided tour of Notre Dame. But visiting the bell towers is not mandatory!
To visit the main floor of the cathedral, the first batch of new time slots is released at midnight (Paris time), for the date 2 days ahead. For example: At midnight (Paris time) on April 1st, time slots are released for April 3rd. Any dates beyond April 3rd will automatically be greyed out/appear to be full. New/additional time slots are then released sporadically up until the day of. However, the reservation system and the release of time slots can be inconsistent, as they continue to improve security measures, so it may not be possible to reserve a time slot in advance.
But you can always visit the main floor of the cathedral without a time slot/reservation by waiting in the "Access without reservations" queue, which is on the left side of Notre Dame (if you're facing it), and is marked by blue signage/banners. The wait time varies, depending on the combination of: the season, the day of the week, the time of day, if there's any liturgical services happening at that time, if there's any special events happening at that time, etc.
For the lowest crowds and the shortest wait times, I recommend visiting before 10:00am on a weekday, or on Thursday evening. Notre Dame opens at 7:50am on weekdays (and at 8:15am on weekends). But for the unforeseeable future, the back half of the cathedral (the ambulatory, the back chapels and the reliquary of the Crown of Thorns) doesn't open until 8:45am on weekdays. For that reason, if you're visiting during the week, I recommend arriving between 8:40am and 9:00am so that you can do the full visitor route of the main floor, while still being early enough to avoid the large crowds!
Or, you're available on Thursday evening, the ideal time to visit is between 8:00pm and 9:00pm. Notre Dame is open until 10:00pm Thursday, everyone has to be out before then, so I recommend entering by 9:00pm at the latest, so that your visit isn't rushed! At 9:30pm, the back half of the cathedral (the ambulatory, the back chapels and the reliquary of the Crown of Thorns) closes, and we start to clear people out of the building.
Yes, guided tours of Notre Dame that highlight the restoration work do exist, but as far as I know, there isn't one that you can join spontaneously.
CASA (Communautés d'Accueil dans les Sites Artistiques/Welcome Communities in Artistic Sites) is a volunteer organization who do free guided tours. They're the "official partner" with cathedral and the Diocèse. Their tours focus more on the religious and spiritual topics of Notre Dame, and less on the architectural and "technical" topics, such as the restoration techniques, etc. I'm not saying this in a negative way! I'm only saying this so you know what to expect.
On weekdays, they offer a combined tour of the interior and of the exterior. On weekends, they offer a tour of the exterior only (due to Notre Dame's restrictions on interior tours on the weekends). The link to their tour schedule is here: https://www.acck.fr/GuidesCasa-calendrierCasa/ (scroll about 1/3 down the page for English). Advanced reservations are not required for individuals. The meeting point is at the base of the north bell tower (but the tour doesn't go to the bell towers), which is on the left side of Notre Dame if you're facing it.
