Just to give a quick update for the benefit of anyone that might face a similar problem. I followed the advise above and I ordered the shaft (gear) from pennine tea and coffee. At the same time, I contacted happydonkey and asked if they do repairs.
I got a reply back that they don't do repairs but it might not be shaft and that it might be the motor. In particular, happy donkey said that if I can't hear any weird noise (which I didn't) it was probably the motor.
The price was very good (around £30) so I decided to order that too. Worst case scenario, I will have a spare motor.
After week I got both with the motor arriving first (happy donkey has awesome service btw).
I changed the motor first and the grinder kept jamming. I got the shaft after that and I changed that too. The grinder works now.
Since I opened the grinder, I decided to replace the motor, replace the burrs with the new that the grinder was sold with and also replace the shaft.
The grinder works great right now! I also managed to dial it pretty fast so happy ending after all.
Thanks for all the replies.