You can upgrade to the BA Premium Amex simply by making a new application, which will be effected as a swap. You won’t get the new card bonus avios, but you can apply/switch. Your card anniversary carries over.

Or is your travel companion your partner/spouse. In which case refer them from you BA Amex to get their own BA Premium Amex. You can then get going on the qualifying spend for the voucher you need right away. However now I’m wondering if you can use your GUFs and Joker on a booking made in your partner’s name. Oddly I think you can, but hopefully a GGL can confirm.

Actually, the middle ground option is to cancel you BA Amex, wait two weeks, then apply for your Premium. You’ll then not have to wait on your next voucher.