As others have said - this will have no influence with regards to entering core surgical training. Having been to a medical which does full cadaveric dissection, it can be a mixed bag and probably is overhyped with prosections being underhyped. Often there's large groups allocated to a cadaver and obviously only one or two people can be actively dissecting at any one time. Initially (when it's all new and exciting, everybody wants to do it) and when there's lots of keen beans, you'll rotate between who dissects while others watch. Then, often, there is usually just a small handful who are super keen on dissection and they become the regular ones in the group to do the dissecting while others watch / chat. Dissecting a cadaver is really different to actually making incisions on a live person so I wouldn't say that gives you any advantage whatsoever. Additionally, this is usually done in pre-clinical (or early years in integrated courses) and it's years before you are a junior in an OR. There's also a tendency for many cadaveric specimens to be suboptimal for learning because amateurs have done the dissecting or that there's been limited time or the cadaver itself may not be the best example - this is where prosection can be markedly advantageous, where somebody has generally spent a lot more time and precision in constructing the specimen with thoughts as to how best to demonstrate the anatomy.
Regardless, you'll need to learn your anatomy to pass your exams. Some medical schools require you to know your gross anatomy in a lot more depth than others, which may just focus on emphasising clinically important points. This may pay off later on with MRCS exams (should anybody some how remember that level of mindnumbing detail). But generally, if you are considering surgery, the best thing to do would be to learn it in depth, including know relations, blood supplies/innervation +/- lymphatic drainage (lolz).
You could then consider intercalating in a BSc like surgical sciences or anatomy or developmental biology. Some of these BScs may have a module or two on dissection or 'advanced dissection' where you painstakingly prepare a specimen.
There's also likely to be plenty of opportunities to do anatomy / radiology / surgical themed SSCs in your pre-clinical and clinical years too which can further give you a boost in knoweldge / enthusiasm for the topic. Many of which does give you further exposure to anatomical specimens.
But going back to your original qn, not doing dissection will have little impact on you being a surgeon.