Freebmd has a birth registered in the March quarter of 1857 in West Bromwich (which covers Wednesbury) for a Samuel Bates 6b 661. He appears to be a likely candidate, however you would still need to know the name of your Samuel's father to know if this is the correct birth.
Surprisingly there are very few Samuel Bates born in Staffordshire around this time.
Let me know if you need help ordering certificates. For NSW certificates it is cheaper to use a transcription service rather than purchase a certified copy. I can point you in the right direction if you wish to go ahead with this.
I suspect you may already know a little more about Samuel than you have revealed in your first post - eg. Where did you get the information about his place of birth? His military service? The information about him being a drummer boy?
The more information you can give the better to save others from duplicating what you already know about Samuel.
