By "emissions testing" if you are referring to local periodic emission testing of the type necessary to renew your license plate sticker and the like, those tests pretty much only rely on the ECU not having any stored fault codes. These cars were designed to not display fault codes for high emissions even though perhaps they should have
You will be fine as long as your "check engine" warning lamp is not on. In other words - for the end user - who was not at fault for creating this situation! - business as usual.
With (I believe) 5 exceptions, as part of the deal for accepting compensation $ from VW, US states agreed to allow these cars to continue to be on the road regardless of whether they had a "fix" or not, and that means passing or failing standard local OBD-based diagnostic tests as per standard procedure, even though the cars are emitting more than they should (which, once again, isn't the end user's fault).
I don't know what's going to happen if you are in one of the 5 states that didn't accept compensation $ from VW.