As an American from the Midwest, I would never use the verb "wish" in regards to objects. I might say, "I would like/love a white wheel." I would use wish more to describe preferable situations or conditions, like "I wish it would snow right before Christmas," or "I wish I had more time."
As a fellow American Midwesterner, I can see where you're coming from, but there are certainly cases where we would wish for objects. "I blow out the candles on my birthday cake, and I wish for a new bicycle", or "I have six apples, but I really wish I had an orange", or "Genie, I wish for a new car!". It doesn't come up terribly often, but it's a perfectly cogent sentence.
When I heard this, I thought the robot voice said white Christmas because jul is probably the most commonly used Swedish word in Minnesota. Since Christmas is such a big deal, we often see JUL written in ads and on banners. I'm so glad you assured us these two words are pronounced exactly the same. Thank you! Now I'm hearing the song "I'm dreaming of a White Christmas, just like the ones I used to know..."
Also, this pun must only work in writing then since the context sure sounds like 'wishing for a white Christmas'.
But it really reinforces that wordplay does not translate well at all, which is something that a lot of people learning foreign languages tend to have significant trouble wrapping their heads around.
And honestly, this one is not bad, you still get most of the sentence the same in a direct translation, so all you need to know is that 'wheel' and 'Yule' are homophones in Swedish to get the joke. A lot of times, there's more involved that makes it far more difficult to understand without a lot more context (good example, 'Have you ever made bread before?' in Japanese is a popular schoolyard pun because moving the first word boundary by only one mora to the right produces 'Have you ever eaten underpants before?').