As with most other things in 5th Edition, this falls under DM fiat.
Does the druid's Wild Shape feature take effort to sustain?
The rules for Resting state that a character must do "...nothing more strenuous than eating, drinking, reading, and tending wounds." (Player's Handbook, 186)
So it is up to the DM to decide just how stressful being a beast actually is.
In my opinion…
Wild Shape ends after X of hours (X = 1/2 druid level) per number of 'uses' you consume (consuming one each instance the time runs out), so it cannot be maintained indefinitely. It also automatically ends if you are unconscious, drop to 0 HP, or die. So, on some level, the character must be focused on maintaining the form and is expending some reserve of energy to do so. As well, once the character reaches the 20th level of Druid Wild Shape has unlimited uses, suggesting it is (finally) completely effortless for the character to maintain. (And by then the question is moot. They can rest in their alternate form but regaining transformations isn't an issue.)
In the end, I would rule against the druid regaining uses of Wild Shape while in their alternate form because they aren't truly resting; it is too stressful on the druid's body. Both from a balance perspective, and to provide a little extra validation to the 20th level Druid feature.