The only way to get Budew in Gen 7 is through breeding, meaning that the odds of it being shiny are either (1) full odds, 1/4096, (2) charm odds, ~1/1365, (3) Masuda odds, ~1/683, or (4) Masuda + charm odds, ~1/512. For this answer, I'm only going to be looking at the absolute highest and lowest chances, so we're only interested in full odds and Masuda + charm odds.
Pokérus has a 3/65,536 (~1/21,845) chance of spawning on a random Pokémon in your party after any battle. But, as Terlor mentioned, it isn't nearly as rare as the numbers imply, since it's really easy to spread once you have it, and really easy to get it through GTS. Even if it didn't spread, it would be impossible to calculate since, for all we know, the Pokémon could have been in 1 battle or 1,000,000 battles and just not participated (plus, the game doesn't decide whether to generate Pokérus after receiving / hatching eggs). But that's no fun, so for our absolute worst-case scenario, let's assume that the Budew has been in the party for exactly 1 battle and just so happened to be the Pokémon the Pokérus was spawned onto (rather than another party member).
Masuda Method and Shiny Charm (~1/512) + infected by a party member (1/1): ~1/512
Full odds (1/4092) + Pokérus spawned specifically on Budew (1/6) after 1 battle (~1/21,845): ~1/536,862,720
Keep in mind that the absolute worst-case scenario is basically just numbers for the sake of numbers. 99% of Pokémon that catch Pokérus do so from a party member.