where I believe that she has met with considerable success
Solution 1:
She is now the head of a private school at Walsall, where I believe that she has met with considerable success.
Like a great many sentences in English, this is syntactically ambiguous: the relative clause "where I believe that she has met with considerable success" could be parsed as attaching either to "Walsall" or to the larger noun phrase "a private school at Walsall".
So it could indeed be taken as saying that she met with considerable success at Walsall. And I don't see anything in the grammar of the sentence that strictly specifies whether she became the head of the private school before or after meeting success.
But in practice, I would say there is little ambiguity, because the sentence makes more sense if it means that she met with success after becoming the head of the school.
Solution 2:
There could be ambiguity, but even with the sentence out of context most readers would probably interpret it as meaning that she has had considerable success since becoming the head of a private school. This is what I think Doyle meant.
"Always" is a troublesome word, and you're right that it could possibly mean she had had success since arriving in Walsall, or that she had had success since joining that particular school in another capacity, and part of that success was becoming the head. I don't think Doyle means either of those here.
The problem is that the present tense says only what is true now, and gives no indication of how long it may have been true. She is now the head of a private school, but she might have been in that role for five years. Present tense might imply a recent development (she is now, [but wasn't two years ago]), but it doesn't specifically define one.
In context, as a summary of Violet Hunter's activities since the case (and particularly as the last line of the story), we already know she used to work as a governess. As no mention has been made of anything else she might have done at the private school or in Walsall, it's reasonable to assume her success has been in the only position that Doyle (writing in character as John Watson) has told us she's held since the events he's just narrated.