Ideally, bullet injuries should be debrided (dead tissue removed), and foreign bodies removed as soon as possible after wounding. However, if a surgeon is not available, or if removal of the bullet/shrapnel would be dangerous or require extensive surgery, it is quite possible for the wound to heal, and for the body to sort of "encapsulate" the foreign body without any adverse effects. If the retained fragments serve as a nidus of infection, and the wound continues to drain pus (be infected), later surgery would be desirable to remove them and aid in healing. In the absence of continuing problems, a retained fragment does not need to be removed. Even today it is not uncommon to encounter patients with retained bullets and fragments without any adverse effects, and these are frequently found accidentally on X-Rays taken for other reasons. The NATO Emergency War Surgery Handbook specifically notes that only readily removable fragments (or others which are more dangerous due to their location--e.g. perhaps impinging on a major artery) need to be removed during initial surgery, and that attempting to remove ALL residual fragments could be more damaging than leaving them in place. In the case of Victorian era British surgery, this knowledge was already in place as a result of experiences in colonial wars, so I don't find the fact that Watson had retained a bullet too surprising-- it is quite possible that a low-energy jezail bullet striking his arm could have healed over before he reached a surgeon, and that subsequently no surgeon felt it was necessary to remove it. He would be very lucky to not have suffered from an infection, requiring later surgery, but this is certainly not unheard of.

I'm not a Victorian, but have done some surgery. Even today we don't always bother to remove bullets. They tend to do their damage in the way in, but much less harm just sitting there. On the other hand, attempts to remove them can do considerable damage. Bullets often fragment as they enter the body scattering bits of metal through the tissue. As you can imagine cutting these fragments out would be time-consuming and difficult, always running the risk of causing further damage to important tissues (Watson mentioned the bullet in his shoulder shattered the bone and grazed his subclavian artery).