英語が上手になりたい is the correct way to say it, and I have always considered 英語を上手になりたい incorrect. I occasionally see/hear the を version from native speakers, but it didn't bother me that much, thinking that this is a mistake that native speakers may also make. But I have been seeing and hearing more of that lately. Today, on two separate occasions two native speakers (mis)corrected beginner learners' ~が上手になりたい to ~を上手になりたい, and I finally flipped the table and came here to put this as a question to the community.
The Internet seems to abound with such examples:
アメリカ人のように英語を上手になりたいので練習(勉強)するとします。(source)
漠然と英語を上手になりたい、と思っていた時と違って何をいつまでにしないといけないかが分かるので、出来そうな気がしますよね。(source)