Do you know of any actual sources (academic or folk) that reference wrapping willow branches around milk/butter containers? I've seen it in several books but none actually list a source and I've not been able to find anything to really back it up?
The Use of Willow in Magical Protection
Do those books say for what purpose the willow is placed there?
Willow or Sally, as it's often called, could be used to make churns so there are many references to that but I'm assuming there is a magical component you're looking for. Most refer to a gad of willow.
I can find a reference to willow and butter together for containers in folklore on duchas.ie in the schools collection, this one is from Galway (link). It seems to be preventive here.
"On Hallow Eve the fairies have power to extract the following year's butter form the churn. To prevent this the churn is encircled by a "gad", a willow rod, which is put on so that the lid cannot be lifted off."
I've also found this being used offensively in form of witchcraft to steal someone's good butter (link)
"In olden times witchcraft was practised very much by evil doers. One of these practices was to take the butter off the milk. The person who was guilty of these acts would go out on a Summer's morning at the break of day and walk in and out through the neighbouring farmer's cattle and take a strug of milk from each milch cow into a cup or vessel and bring it home to her own house. She would be usually about to churn when this would take place. She would then start her black magic. She would get a gad and tie it around the churn dash which would then be fixed and ready to churn her own cows milk. It was said that the whole magic lay in the gad that was around the churn dash. The gad was made of young sally rods. Then the churning would proceed and heaps and heaps of butter would rise on and in the churn with the result that before the churning was competed portion of the butter would have to be removed to make room for the remainder of the butter to come on the churn."
While not specified to be willow I do find more references to fairies, gads and Lá bealtaine (link)
"Long ago when the people churned on May Day they would put a "caoráin gad" round the churn to prevent the good people from bringing the butter"
What is a gad? It seems to be a stick or more of flexible sticks stripped of their bark, crushed and made and twisted together then possibly into a circle. (Link)
"The round berry gad was put on the top of the lid. Here is how the round berry gad was made. On May eve the people went out and cut the round berry rods. Then the people peeled the rods and then they twisted the rods and made a gad out of them"
"made of rods crushed and twisted to form what is called a "gad""
Though I did also find this one from Wexford which is unrelated but implies a connection between dairy and willow (link)
"Long ago when old people were putting out their cows, they hit each cow with a willow stick between the horns. The reason was that nobody would be able to take their butter from them then"
While gads seem to typically be willow I did find one for Rowan as well (link)
"Some people as well as going through all the other doings which I have mentioned, put "gads" of rowan tree up over their doors and they also supposed to prevent the losing of the butter."
I hope this was in anyway helpful. This seems to be reasonably well attested to. Willow seems to be used on dairy containers to prevent the Othercrowd from taking it and can also be used to steal it. This seems to be a practice year round but done especially at Lá Bealtaine but also in some places on OĂche Shamhna.
Thanks for asking, I'm always happy to help!