I don't think any team will overtake NZ in the rankings in the next decade (given the youth of the Chiefs talent etc), but plenty could knock them out of a RWC or a one off test. As Eddie Jones points out, its not just playing talent, its coaching talent and the fact these coaches learn across the world and bring that knowledge back to NZ.
http://www.rugbyworld.com/countries/new-zealand-countries/do-nz-run-world-rugby-69892
Focussing on rankings, England are closest to them in the rankings, but the 6N sides all generally have lower rankings than the RC teams, so it makes it harder for England (or any NH team) to close the gap, particularly if they only play NZ once in a blue moon as is the case at the moment.
- England would be my first choice if they can avoid the French trap (or English football trap) of becoming slaves to their clubs and clubs focus on generating their own youth talent rather than buying their way out of trouble.
- South Africa would be my second choice if the expansion of rugby to different demographics of their population turns out to be genuine rather than imposed on them. If residency rules are extended and "project players" becomes a thing of the past then I'd make South Africa more likely than any NH side to challenge NZ.
- Argentina third if they get another SR franchise or two, improve their coaching at club level and the other Americas teams increase their level enough to really compete with an Argentine second XV. They'd need a replacement as good as Hourcade though.
- Ireland fourth as they are heading in the right direction and are slightly ahead of Wales in terms of development I think and are twice the population. Again, they'd need a replacement as good as Schmidt.
- Just to give you all a laugh, but my fifth choice to be the next side to overtake NZ in the rankings could be Japan. But that wouldn't be before RWC 2027! It'd depend on their RWC being a success, more SR franchises, continued investment in the highest quality international coach available, inclusion in the RC and a strong commitment to have the Top League be a feeder league for SR. But I think all those things are achievable and that the JRU are intent on achieving them. Only 12 places and 21 ranking points to go!
All the above could potentially do it, but I can't see them knocking NZ off the top in the next five to ten years unless the format of the 6N and RC changes dramatically so there is potential for an annual NZ vs England match (as Hansen has called for). I think they'll be the worlds top five rugby powers by this time next decade (only three of them are currently in that top 5). I think Australia and Scotland will decline and Wales and France will stall. All completely hypothetical though.