Thank you very much!I might simply say "His English speaking skills improved to intermediate" or "to intermediate level" or "to the intermediate level," depending on how you wanted to end the sentence.
"Improve up" strikes me as redundant since we normally think of improvement as an upward movement.
Thank you very much!It should be "His speaking skills have improved up to the intermediate English level."
This means, "In the earlier levels, his speaking skills improved." However, it comes with the implication that his speaking skills did not improve in the intermediate level.
I am not sure that this is what you mean.