Under EASA regs coming into force in April with an implementation date as yet to be confirmed but not long after you will be required to hold dual licences and medicals in order to operate a non EASA aircraft in EASA territory.
That was extended from 2012 to 2014 by EASA with a further extension likely to 2016 while the Bi Lateral agreement with the USA which has already been signed could be further agreed re FCL. As EASA stated dual licences is not their preferred route!
So maybe not quite "soon after unless you clasify soon after as 2-4 years
Regarding ATP and CPL pilots already working in Europe the proposed regulations already run against existing EU employment laws which EASA are aware of hence the silence re that category of licence.