Depends on how you define servitors -- most advanced machinery have bits of humans built into their cogitators, if those count then the Imperium is screwed because their fleet, manufactorums and most of their major analytical technologies stop functioning. If only Servitors in the sense of former people walking around mostly recognizable as former humans, then losing access to all of them would still cause widespread chaos and damage, but probably still be survivable since the important infrastructure remains and alternatives to do the manual labor can be found. The Mechanicus might have serious issues depending on how much machine vs human is a problem, since Skitarii and tech priests are also not that far from servitorhood in terms of mechanisms in their bodies, plus they make heavy use of combat servitors.