I know that sharp key signatures are applied in the order: F C G D A E B and that flat key signatures are applied as: B E A D G C F. But these are for major and minor scales, right? Is there a way to represent harmonic or melodic minor scales with the key signatures?
The short answer is: no there isn't. There's only a key signature for the natural minor. When the harmonic or melodic minor scales are used the seventh or sixth scale degree notes are sharpened by adding an accidental.
It's useful to think of the melodic and harmonic minor scales as ad hoc adaptations of the natural minor scale: Minor pieces just use the natural minor scale. But when the seventh degree note is used in a harmonic sense (to form a chord) it is temporarily sharpened to make it a leading tone. And when the sixth, seventh and eight degree are used in succession (melodically) the sixth and seventh are temporarily sharpened, to create a leading tone of the seventh and to make the space between the sixth and the seventh less jarring.
No because harmonic minor is not a key it's a scale. Key signatures are only for designating keys not scales. With a key you are talking about the idea establishing a tonal center within a structured harmony while a scale is just a set of notes. You don't need to stay strictly in a scale to establish a key and just because you use notes in a scale does not mean you establish that as a key. There's actually an interesting composition technique called Pandiatonicism where you use the notes of the diatonic scale to avoid functional harmony.
The harmonic and melodic scales are derived from manipulating the minor scale for melodic and harmonic purposes, but the base key is still natural not harmonic or melodic. You can see this answer for more specifics on how this is done.