How are NATs connected to European Airways?
There is no equivalent in Europe to the North Atlantic Routes, you simply file direct from point to point.
This is the routing for a flight on Wednesday 18th of January, 2017 (DAL4).

KJFK
MERIT HFD PUT WITCH ALLEX N263A JOOPY NATS RESNO NATS NETKI NIBOG REMSI UL603 SOGPO UT149 OTSOP T149 LIPMI T150 ROLIS
EDDF


You see that the route terminates NAT S at NETKI NIBOG REMSI which are directs, until the flight joins the upper airway UL603 at REMSI.

Here is the routing in SkyVector.

And here is the screenshot from the FlightAware flight.


(Image Source: FlightAware)