Your original suggestion of "recover your senses" sounds rather formal/poetic, but it's definitely understandable. I wouldn't use it in casual conversation, but I'd get it if I heard it.
Before I get to an alternative phrasing, there are a few other problems with your sentence. You wrote:
How long a time does it usually take before you "recover your senses" in the morning after rising from a night's sleep?
I've bolded the areas of concern. First, we don't say "how long a time". "How long" in this context already implies that you're talking about time. When inquiring about event duration, we simply say "how long does it take". For example:
How long does it take to bake cookies?
"After rising from a night's sleep" is very awkward; no one would say this. In addition to being awkward, it's unnecessary; in contexts like these, "in the morning" is already sufficient enough to imply "when you wake up in the morning". If you wanted to, you could write out the full "when you wake up in the morning", but "in the morning" says all you need it to.
So taking these changes into account, we're left with this:
How long does it usually take before you "recover your senses" (when you wake up) in the morning?
Perhaps a more idiomatic way to refer to "recovering your senses" would be "to fully wake up". We use "fully" to mean that, beyond simply being awake, someone is completely aware and is no longer sleepy/still waking up. I think this is probably the way you're most likely to hear a native speaker phrase it, and since "before you fully wake up when you wake up in the morning" is very redundant, I'd leave out the "when you wake up" and simply say:
How long does it usually take before you fully wake up in the morning?
Best Answer
There is nothing wrong with your phrase
it is understandable and grammatically correct. Metaphorically it has implied "movement" which is characteristic of progress. All native speakers would understand it.
An alternative with similar meaning might be