If it is booked under the BD flight No, this is a BD L class (discounted new economy) not an AC L class (discount economy) and the voucher is definitely valid for upgrade from new economy to the business.

Try at EDI first. They might conceivably upg you right away if there is plenty of availability at that stage. If they can't upgrade you straight away, they should put you on the ugrade list and the sooner you are on it, the greater the chance of upgrade. No need to re-check in at MAN (check all the way throughtat EDI) but ask again at lounge and/or gate if upgrade has cleared.