In theory, if it is booked with a BA prefix then you should be granted lounge access. In reality, I can foresee another confusing situation for all concerned, with the old "Vuelling is only a partner and not a Oneworld airline" quotes.

Under the current access rules, I don't get why being a gold or silver would make an iota of difference if only trying to access a business lounge. I could however understand if this answer was some years ago when gold's had the benefit of lounge access when travelling on any airline.