Sticking with just the Premier League, there have been five other game weeks in which three hat-tricks were scored:
- 23 September 1995, when Robbie Fowler (Liverpool), Alan Shearer (Blackburn), and Tony Yeboah (Leeds Utd) scored hat-tricks
- 27-28 September 2003, when Ruud van Nistelrooy (Man Utd), Kevin Lisbie (Charlton), and Steve Watson (Everton) scored hat-tricks
- 25-26 October 2019, when Jamie Vardy, Ayoze Pérez (both Leicester), and Christian Pulisic (Chelsea) scored hat-tricks
- 23-24 October 2021, when Mason Mount (Chelsea), Joshua King (Watford), and Mohamed Salah (Liverpool) scored hat-tricks
- 1-2 October 2022, when Erling Haaland, Phil Foden (both Man City), and Leandro Trossard (Brighton) scored hat-tricks.
Fascinating that they all occurred in September or October...
I suspect that if you go back to the Division One era, when matches tended to be higher-scoring, you'll find game weeks with larger numbers of hat-tricks, but there don't seem to be any convenient resources for finding hat-tricks scored during that era.