How do you pronounce "Weeoanwhisker"?

Given it's London location, I'd opt for a British pronunciation, something very like /wiːəʊnwɪskəz/ --- "we own wiskaz". (While I personally disapprove of non-rhotacism, I get that Londoners and the English in general do insist upon it.)

I would guess it's supposed to be a pun on "we own whiskers" (whiskers being another word for beard). So just pronounce it like you would that.

JKR loves wordplay, and her other shops have vaguely pun-like names related to what they sell

• Gringotts: Bank. References ingots (bars of gold)

• Flourish and Blotts: Bookshop/stationers. Both things you can do when writing in ink (although hopefully the first, rather than the second!)

• Twilfitt And Tatting's: Robes. Twill is a material, clothes are being fitted there, and tatting is lacemaking.

• Eeylops: Owl emporium. Could be from a combination of Eagle and Scops owl.

• Gambol and Japes. Joke shop. To gambol is to spring around happily, and jape is another word for joke.

so when other names were made up for the films or attractions whoever came up with them might have followed this convention.

This seems likely, as in the films/games/Wizarding World of Harry Potter we also have

• Slug and Jigger: Apothecary. Slug and jigger are both measurements of liquid (and horned slugs are a potion ingredient)
• Amanuensis Quills: An amanuensis is somebody who takes dictation, so would need a writing implement.
• Rosa Lee's Teabag: Café. Rosie Lee is cockney rhyming slang for tea.