Obviously as the BT Mod has offered an invite to contact them to sort this out , you should be on your way to getting what you want, but it seems to me quite obvious the problem and the solution.
If you had Sky Tv in your name with BT Sport added, (from BT not Sky) and you in effect have cancelled your Sky account (and viewing card) obviously BT Sport won't work as your Sky card is switched off, your wife now has a Sky account and card ( presumably Sky needed to issue a new viewing card to accomplish this)...so you have a BT Sport 'account' on a Sky card that is no longer valid and need BT Sport added onto a valid Sky viewing card ( the one in your wife's name)....TBH , if you have just advised BT of this change how could you expect BT to know, and if you arranged this with Sky , perhaps Sky should have told you what to do to ensure you didn't lose BT Sport in the changeover period.
You will probably have to cancel your BT Sport account with BT, and your wife will have to ask BT to add BT Sport to her Sky 'card' ...if you are paying for BT Sport as a standalone product that should be OK, apart from the fact that there may have to be a notice period to serve, but if 'your' BT Sport was free or reduced as part of having some other BT product, then the mis match between presumably your name on the BT account and her name on the Sky account could cause problems, if you want it to continue to be free or cheaper.
