Context:
I would reply to your email after 10 days. I would like to appreciate your patience in this regard.
Can I use "aspect" instead of "regard".
I think both will mean same but I need explanation.
The short answer is it's not a good idea to replace regard with aspect in your specific context.
Whichever word you used, it would be a metaphoric extension from the core meaning, which in both cases centres around the concept of a perspective, vantage point, point of view.
Note that [seen] from this aspect is a credible (though less common) alternative to ...in this regard when referring to a specific way of looking at something. But in OP's context we're actually referring to everything about the thing itself, not just some particular way of looking at it.
Idiomatically, only in this regard or in this respect are well-established for the extended meaning concerning this matter.