This is only my opinion, but I would say it is acceptable but still incorrect. It’s definitely less correct than the version with が.

I read the version with が this way.

（私は）［英語が上手］になりたい。

The part I put in brackets describes the desired stated for the speaker.

In contrast, I read the version with を this way.

（私は）［英語］を上手になりたい。

This is incorrect because 上手になる is not supposed to take a direct object to be marked with を. On the other hand, it is acceptable because 上手になる means the same as 上達する and 英語を上達する is somewhat acceptable, although 英語を上達させる should be more correct. 英語 can be seen as an object of deliberate improving.

Unlike a similar sentence with 好き, the following statement would be totally unacceptable.

x（私は）英語を上手だ。

This is because the adjective 上手 by itself doesn’t seem to work towards any object, unlike 好き. It’s more static, so to speak. It is the change indicated by なる that is making を somewhat acceptable.

It doesn’t always work, though, as the following is not acceptable.

x（私は）頭をよくなりたい。

This is only my opinion, but I would say it is acceptable but still incorrect. It’s definitely less correct than the version with が.

I read the version with が this way.

（私は）［英語が上手］になりたい。

The part I put in brackets describes the desired state for the speaker.

In contrast, I read the version with を this way.

（私は）［英語］を上手になりたい。

This is incorrect because 上手になる is not supposed to take a direct object to be marked with を. On the other hand, it is acceptable because 上手になる means the same as 上達する and 英語を上達する is somewhat acceptable, although 英語を上達させる should be more correct. 英語 can be seen as an object of deliberate improving.

Unlike a similar sentence with 好き, the following statement would be totally unacceptable.

x（私は）英語を上手だ。

This is because the adjective 上手 by itself doesn’t seem to work towards any object, unlike 好き. It’s more static, so to speak. It is the change indicated by なる that is making を somewhat acceptable.

It doesn’t always work, though, as the following is not acceptable.

x（私は）頭をよくなりたい。