Yes, if you are connecting both to the internet on the same router then you will likely run into NAT type issues. Even connecting a single system can have these issues because they are Microsoft / Whoever Built Your Router compatibility issues.
In the past I have tried Microsoft's support number and they walked me through opening ports and a bunch of other nonsense to get both my Xbox and my roommate's Xbox working with at least one of us having Open NAT type, however it was never possible for them to get us both Open NAT with at least one of us always being stuck on Moderate or Strict. They said the only solution was to use a different router for each system.
However since you are likely using a single login and one cannot login to two identical systems at once this might not affect you, unless you have both of them on at the same time when you switch systems. Once you turn them on they sort of "lock in" their connection settings. In the example above, whichever of us turned on our system first would "keep" the Open NAT type until we had BOTH switched off our systems. Even if I got it first, he turned his on, and I turned mine off, his would remain on Moderate unless he restarted too.
Basically whichever one you want to have a better connection on, turn that one on first. Or not worry about it by only using the older system as an offline only setup. :)