Kellaway secured a golden-point win over Turkey's Erman Eltemur in the -75kg class, while Connell beat Jelena Maksimovic of Montenegro 4-3 in the -55kg kumite division.
The results boost their chances of qualifying for the 2020 Olympic Games.
"Moments like this are what we do it all for," said Connell.
Kellaway said: "I've been on a bumpy ride but I finally put it together for this moment. To be on a podium with these three other athletes is a dream come true."
Karate will make its Olympic debut at Tokyo next year and by securing bronze medals both fighters have guaranteed themselves places at the 2019 Minsk European Games in June, which will be a key Olympic qualification event.
Congratulations to Amy Connell winning Bronze at the European Senior Championships 2019
Well done any from all at the SKGB