If you wish to have the author’s products listed, you need the post type specification or only regular posts will be listed. Having a page ID as part of the URL parameters would be inappropriate to get to the author’s post list page. I don’t know where UID is coming from, author pages are usually linked as ?author=3
I don’t think WC is the problem. My product author page is linked as example.com/?author=1. This is probably my theme’s doing, it leads to the normal author page listing posts, not products. example.com/?author=1&post_type=product will yield an author page listing his products.
The question I suppose is what function is outputting the author link on your product pages? It is often get_the_author_posts_link(), which normally links with the ?author=3 parameter. If it is anything else, your theme or one of your plugins is altering what is output by hooking the “author_link” filter.
If you cannot resolve the issue through corrected theme or plugin filtering, you too could hook the filter with a large priority argument so your callback is called last. You can then alter the link in any way you want. But only if the link is coming from the get_the_author_posts_link() function.
