If a person did not own a FA or CA prior to the cutoff dates, they can never own one. AND they had to be in possession of at least one firearm of the class as of the implementation date, AND they have to be in continuous possession of at least one specimen in each category to retain grandfathered status.
No one can own an original MG34 in either class unless grandfathered, with a PAL valid for 12(2) or 12(3) as the case may be.
If the firearm is of new manufacture, originally semi-auto, and has passed inspection, then it can be acquired with a current PAL. There are a variety of firearms in this category, with more alll the time. In all likelihood, this new German made MP44 would qualify. I can't imagine that the German authorities are more lenient in their inspection criteria than ours are. The MP44, with its separate safety and selector might be one of the easier designs to manufacture so that it is unquestionably semi-auto. With no provision for a selector or any auto fire components, its semi-auto status should be easy to establish.
Unblocked MP44 mags are prohibited devices. M-1 rifles are specifically exempted, as are belts for a firearm originally designed prior to 1945. The 1945 date is relevant only for belts. This one can be a bit tricky; if you have one of the Browning clones, cloth belts or pull out links as originally used are acceptable; newer designs which can be used as push through or pull out may not be.
Assuming that these new MP44s and MP38s are actually available for purchase, I'm sure it will be a matter of time before someone goes through the process of importing them.