No, they can’t. The rules say:
If your side (but not necessarily your power) controlled a canal or narrow strait at the start of your turn, you may move sea units through it (you can’t use it in the same turn that you capture it).
Related to this, there’s a well-known Allied ploy to have the Americans capture Denmark, and then move the British through the Denmark Straits to capture Germany, which is perfectly legal because the Allies indeed control Denmark at the start of the British turn in that case. But that situation is different from the one you’re describing of course.