AirPrint doesn't work with new wifi router

The HP LaserJet P1102w is not mentioned in About AirPrint - Apple Support, but it was in fact listed as supported in a very old HT4356 iOS: AirPrint 101 article (February13, 2012). It was then possible to use the printer with AirPrint (if necessary, after a firmware update).

Just in order to avoid any misunderstanding, do you mean that you have used Apple's AirPrint technology, not just wireless printing in general (which would have required additional drivers)?

A P1102w would use 802.11 b/g Wi-Fi and 2.4 GHz only. This would be the first thing to check. Make sure that the new Wi-Fi router will accept 802.11 b/g connections at a compatible wireless security level, and that the printer can use a 2.4 GHz band.

Modern dual-band routers may have an automatic/"smart" device detection and band selection, which could cause difficulties with older printers. If so, the router settings may have to be changed, sometimes involving splitting and renaming bands.

https://www.hp.com/hpinfo/newsroom/press_kits/2010/plugandprint/pdf/LaserJetProP1102w_3.30.10.pdf

Seems to me you should be on a forum for the router and not your Mac. However before doing that:

1. Remove the printer from your printer list
2. Restart the Mac in Safe Mode
3. Restart the modem and router and PLEASE ensure each has up-to-date firmware (yes, you will need to read their manuals)
4. Restart the Mac normally
5. Finally re-add the printer and test.

If you configure AirPrint with a printer on one network segment or WiFi SSID and the Mac connects to a different network segment or Wi-Fi SSID, then AirPrint won't find the printer. You would have to either use the original network segment and/or WiFi SSID or remove and add the AirPrint printer on the current network.

As Jan Hedlund has mentioned, there is a question of this now being a supported AirPrint device, or one requiring compatible vendor drivers, which owing to its age, would not be available from HP for macOS Sequoia.