The vast bulk of milk sold is pasteurized. It contains the remains of H5N1 viral DNA -- so we know that it's in the milk supply -- but the pasteurization inactivates it. Unless you're intentionally buying unpasteurized milk, you're probably using pasteurized milk. From what I've read, influenza is very vulnerable to heat, so pasteurization readily affects it.

I doubt that anyone's tried testing unpasteurized milk mixed drinks.

googles

This says that you can inactivate H5N1 with ethanol-based hand sanitizer, but that it's comparatively-resistant to it relative to other forms of influenza, FWIW.

https://wwwnc.cdc.gov/eid/article/28/3/21-1752_article

Evaluating the stability of highly pathogenic avian influenza viruses on human skin and measuring the effectiveness of disinfectants are crucial for preventing contact disease transmission. We constructed an evaluation model using autopsy skin samples and evaluated factors that affect the stability and disinfectant effectiveness for various subtypes. The survival time of the avian influenza A(H5N1) virus on plastic surfaces was ≈26 hours and on skin surfaces ≈4.5 hours, >2.5-fold longer than other subtypes. The effectiveness of a relatively low ethanol concentration (32%–36% wt/wt) against the H5N1 subtype was substantially reduced compared with other subtypes. Moreover, recombinant viruses with the neuraminidase gene of H5N1 survived longer on plastic and skin surfaces than other recombinant viruses and were resistant to ethanol. Our results imply that the H5N1 subtype poses a higher contact transmission risk because of its higher stability and ethanol resistance, which might depend on the neuraminidase protein.

"EA" here is "ethanol alcohol"

All influenza viruses were completely inactivated (below the detection limit) within 15 seconds by treatment with 40%, 60%, or 80% EA or 70% IPA (log reductions in titers were >4). However, all viruses were not inactivated by 20% EA (log reduction <1). Of note, although all subtypes except for H5N1 were completely inactivated within 15 seconds by 36% EA (log reduction >4), the disinfectant effectiveness of 36% EA against H5N1-Ky and H5N1-Eg was substantially low (log reduction <3) (Table 3; Appendix Table 1).

I don't know the percentages for mixing a White Russian offhand. WP's mix for it is:

https://en.wikipedia.org/wiki/White_Russian_(cocktail)

• 5 cl (5 parts) vodka
• 2 cl (2 parts) coffee liqueur
• 3 cl (3 parts) fresh cream

Coffee liqueur doesn't have a fixed ethanol content.

https://www.liquor.com/best-coffee-liqueurs-5086056

On that page, the coffee liqueurs are 20-35% ethanol ABV.

https://en.wikipedia.org/wiki/Vodka

That says that vodka is 35–40% ethanol ABV.

If you go by that recipe and numbers, then a White Russian could range from 21.5% to 27% ethanol. That won't be enough to inactivate H5N1 in 15 seconds, at least based on the above research. I don't know what kind of inactivation time is required to be safe for consumption.