In my opinion BOM should be in the control of Design and Production Planning Control. Since these are the origin departments for products manufactured in the company.
It depends business to business. Production will have to know it, finance will input because of costs and trade will input because it is associated with the item. I would say production first and foremost, but T&L is there as well. “Supposed” is not really a word I would use, it would depend upon the implementation sessions and who has skills where and what the customer wants - no point in discussing BOM’s in T&L if production own them.