There is no pet schedule, you can only sort-of-kind-of predict it.
Well, there’s sort of a pet schedule. Once per week, there will be a 24-hour pet rescue. That one is guaranteed. During that time, any pet gnomes you find will just drop food rather than starting a new rescue.
As there is not a currently active pet rescue, if anyone in your guild encounters a pet gnome and wins the combat, it starts a pet rescue that lasts for 1 hour. Pet gnomes are like normal gnomes and fairly rare. Unlike normal gnomes, they seem to only appear in PvP/Arena.
So if your guild is very active, then pet rescues will start roughly every hour due to guildmates finding them and winning. If your guild is small or inactive, your odds of someone finding a gnome are dramatically lowered.
For example, if a 30-member guild is grinding PvP matches and finishing 2 per minute, that’s 60 chances per minute to find a pet gnome. But if 3 people are taking 4-5 minutes to finish matches, that’s roughly 0.75 chances per minute to find a gnome. If we think the odds of one displaying are about 1/30, that means the big guild has a very good chance of finding a new pet gnome within 5 minutes, but the small guild might take several hours to find one.