When do pilots have to submit position reports?
It's not fixed. I checked ICAO Annex 10 Vol II that lists the SARPs for communication. It will be down to the FIR the flight is being conducted in. For the NATS (UK) side of the Atlantic, I found this document (may not be current).

The basic intervals are whichever comes first:


Waypoint on the flight plan
One hour had passed
ETA has changed for the next waypoint by +3 minutes
Interval based on whether north or south of 70°N.


CPDLC is used, and can be automated too (ADS-C). If the FIR does not support CPDLC, then it will be via voice. According to Boeing, FANS 1—which handles reporting via datalink—saves a lot of time and improves accuracy:


  The uncertainties of traditional voice position reporting and the delay associated with high-frequency relayed voice communications [result in 20 to 45 min to make a position report].


And depending on the ADS-C (C for Contract) agreement, the reports can be as frequent as every 5 minutes.



For the US side of things, check section 5-3-2 of the AIM and the section titled additional reports. It's similar to the above. They also note:


  For aircraft providing automatic position reporting via an Automatic Dependent Surveillance-Contract (ADS-C) logon, pilots should discontinue voice position reports.