As a fast bowler, if you take 600+ wickets at an average of 26 odd, you are one of the all-time greats without any doubt. It does not matter what your averages are in in individual countries.
Besides, he has bowled memorable spells in pretty much every country.
Anderson, Cook, Pietersen, Stokes and Root are the five all-time great Test cricketers England have produced in this millennium.
By those standards for this millenium:
Australia:
Hayden, Ponting, Clarke, Smith, Gilchrist, Warne, McGrath
India:
VVS, Sachin, Dravid, Kohli
New Zealand: Williamson
SL: Murali, Sangakkara, Jayawardene
SA: Smith, Kallis, Steyn, Philander, De Villiers, Amla
Looking at cricketers who have played a large portion of their careers this century, all who I would consider players of the modern era
For me, 25 cricketers is too many in a single era to be ATG's
The test ATG's debuting post 2000 would be S.Smith, Steyn, Kohli, Sangakkara, De Villiers. Current established players with the potential to become ATG's are:
Cummins, Bumrah if they continue their current records
Rabada if he improves his overseas record
Williamson if he gives major showings in Ind, Aus
Root if he can finish with a 50+ average and 30+ centuries (that number should be easy for someone who players so many tests. If Smith/Kane.Kohli had played so many matches they would have got there/be close)
Stokes if he can continue his form over the last few years
KP was too inconsistent and Anderson, Cook fall just short of the dominance over the majority of conditions required to be an ATG. Instead, they fall in the ATVG tier with the likes of Hayden, Walsh, Kumble etc
clouderson 
