It could mean either, although it probably means the latter, that not all of us were happy with the result.

For it to mean that everyone was not happy with the result, you would have to assume that the speaker omitted the word "that":

We weren't all [that] happy with the result.

Or, that the word all is misplaced:

We all weren't happy with the result.

But it's probably better not to assume anything and conclude it's the latter.

It could mean either, although it probably means the latter, that not all of us were happy with the result.

For it to mean that everyone was not happy with the result, you would have to assume that either:

(1) The word all is misplaced:

We all weren't happy with the result.

or (2) The speaker omitted the word "that":

We weren't all [that] happy with the result.

But it's better not to assume anything and conclude it's the latter.