I think that showing users how long they need to wait to ask a new question is a good idea for a few reasons.
Consistency with other rate-limit notifications
According to The Complete Rate-Limiting Guide, nearly every rate-limit notification lets the user know how long they've been restricted from a specific action.
Time-specific notifications are shown for voting, comments, flagging, chat bookmarks, and accepting answers. I don't see a reason to exclude them for asking questions.
Rate-limiting applies network-wide
With permission from site moderators(Workplace, Academia, ELU, Arduino, Code Golf), I have been posting meta-questions across multiple sites to test if YouTube embedding was enabled.
I initially came across the rate-limit while trying to test Android Enthusiasts. When I hit the limit, I checked my userpage as per Adam's advice, but my only question on the site was from over a month ago.
Later I realized that my rate-limit was from a post I had made 30 minutes ago on Arduino.
Fun fact: Stack Exchange gets very suspicious when cross-posting a question with multiple external URLs and HTML tags across multiple sites in a short timespan.
As Adam noted, if a user really needs to know how long they need to wait again, they can simply look at their user profile.
If that's the case, showing how long users need to wait isn't giving away any information they wouldn't already have access to. It's simply making the information a bit more conveniently accessible.
TL;DR: While this isn't a critical feature by any means, it would be a nice quality of life improvement. I don't see any reason not to add it.