BA don't have a First service to VIE, just a Club Europe service, so I presume it's on the longhaul? In which case it would need to be a BA or other oneworld airline. But if it is, then you can use BA First, Cathay First, Qantas (no First section, but it's brand new) and AA Flagship, as well as their business class equivalents.

More information is available in the Lounge Eligibility thread in the Dashboard and also this recent thread:
https://www.flyertalk.com/forum/brit...e-options.html