I don't really think so. Bird flu doesn't spread super easily from person to person as it usually is and not a whole lot of people breathe in vast amounts of goose shyt and spit every day (I mean, ducks, chickens, etc...but you get it). I guess it could if it mutated in a way that made it more contagious between humans. Another reason pandemics happen is because a virus has a relatively low case fatality rate. Bird flu's CFR is sitting pretty at 54% in humans, and sometimes birds with it will die as quickly as 48 hours after getting it, so it doesn't have ideal characteristics to spread to a whole lotta people. Meanwhile the Beta variant (highest CFR) of covid19's CFR is less than 5%. It's possible, anything is really, but I just don't think bird flu is something people that don't work with poultry, livestock, or waterfowl should be worried about right now. Just wash your hands right after touching dead stuff.

Mpox has been a PHEIC since August and spread to the UK recently though. Needs bodily contact for transmission so even if that becomes a major issue we're not going to have to keep everyone at hockey stick length again.