Sort of, but not as well as you might like.
The first thing to understand is that Warlock Pact Magic doesn't stack with other kinds of casting. If you have a character who has 3 levels in druid, and 3 levels in cleric, they have the spell slots of a 6th level character. (Their spells known only go up to spell level 2, but they can upcast.) if you have someone who has 3 levels in druid and 3 levels in warlock, they will cast pact magic with the slots of a 3rd-level warlock. They will also cast druid magic with the slots of a 3rd-level druid. If they've cast out of pact magic slots, and they short rest, they get back all of those pact magic slots, as warlocks do. If they've cast out of druid slots and they short rest, they can use Natural Recovery to recover some of those slots... as long as they haven't exhausted their ability to use Natural Recovery for the day. However, their ability to recover druid slots on a short rest doesn't benefit in any way from their warlock levels.
It is the case that as a multiclassed druid/warlock, you can cast spells off your druid list using your pact magic slots (which you can then recover on a short rest) as long as the level of your pact magic is high enough to cast the spells in question. That has nothing to do with Natural Recovery, however. Likewise, you can cast warlock spells off of druid slots, if you wish to do so.
As a further warning, this character is likely to be frustratingly weak, compared to party members who pick a casting class (or non-casting class) and stick with it. in 5th ed, multiclassing is usually a mistake, unless you can find a combination that works particularly well together for some reason. In the case of Druid/Warlock it's particularly bad because the druid craves Wisdom and doesn't care about Charisma, while the warlock craves Charisma and doesn't care about wisdom. If you want a caster who runs on short rests, play a warlock. Possibly talk with your DM about adjusting the spell selection, to allow more druidic spells via pact magic, and go pact of the tome so you can pick up some druid cantrips. If you want to play a druid, play a druid. The Natural Recovery of a pure druid will be much more effective than the Natural Recovery of a Druid/Warlock multiclass. There will be more druid levels to power it, after all. If you want to play a druid that mostly runs on short rest resources, consider playing a Circle of the Moon druid, and focusing on shapeshifting. Natural recovery is nice, but it isn't that big a deal. It's basically one more casting of whatever your highest level spell slot is.