I’ve been using a 16 kg kettlebell for about 6 weeks now, alongside daily swimming and exercise bike sessions. I’ve lost 11 lbs, feel healthier, fitter, and can feel a real difference in strength in my upper body already. What I’m wanting to do now though is bulk-up quite a bit, especially in my shoulders, chest and arms. If I move up to a 24kg kettlebell will I be able to develop a noticeably bulky muscle mass over the next year if I train hard 3 times a week? Is this weight heavy enough to do this?

Yes you at "First" will gain a bit of bulk. but your body will eventualy adapt to the 24kg kettlebell. Progressive Overload is the key to developing in the gym.

Bulk="Eat big. Lift Big. Be Big!