I don't work in web development but accelerating CCL algorithms do happen to be my area of expertise. While their execution time depends on the image content, you can expect them to be in the tens of milliseconds for a 4K image, at least for the 'modern' one you can find in OpenCV, and on current consumer-level CPUs, without multi-threading.
Of course, implementing these newer algorithms isn't straightforward. That's why you should instead use something like OpenCV if you have the option to.
Moreover, if CCL doesn't fit what you're looking for then you can also check out watershed algorithms ( https://en.wikipedia.org/wiki/Watershed_%28image_processing%... ). Because they don't work on binary image, they might be a better way to describe what you call 'enclosed space'.
There are faster fill algorithms than the above, described on Wikipedia, that don't need you to visit the whole image [1]. In particular, span-based filling.
You'd first threshold the image by checking if a pixel is 'close enough' to the clicked pixel. This will create a B&W image. Then you call the CCL on this binary image to produce a labelled image: each pixel will contain the id of its component (i.e. which enclosed space it belongs to). Once you have this, it's just a matter of replacing colors on the canvas for pixels with the same id as the clicked pixels.
Obviously, that's just the 'theory' part. If you can find a good library that includes one then you'll have to integrate it. Otherwise, you'll have to re-implement that in Javascript, which is easier said than done and may also not reach the desired performance.