To answer your question , she won’t have any problems. I have the same situation. My Thai passport has a different name and my Australian has a different name. The Australian immigration links them together in the data base . If you are unsure she can request them to do next time she checks in at the airport in Australia.
Simple, it’s the same as everyday before the outbreak. The farce of downloading an app and be on your merry way. This was the case last night (Wednesday)
you only need your birth certificate with the Thai parent on it and the house book that your mother is registered on. It’s quite simple and of course your current passport. All these documents must be translated into Thai except the Thai house book.