But since you are 2 junglers you gank the lanes all the time, just let bot play deffensively.
Use blitzcrank and alistar, it wins games. In fact it did even before the changes.

I can see this working with pre-made teams, but not "on the fly".

You need a champion that is viable 1v2 bottom, and a viable 1v1 top and mid. Now, to be fair, 75% of champions or more are 1v1 viable, with it all coming down to skill and playstyle.
However, only 5-15 champs can really handle their own 1v2 assuming equal skill level.

Why do I say that?
In order to 1v2 successfully, you have to be able to:
1) Defend yourself under the tower when enemy creeps deflect initial aggro.
2) Recover from heavy harass.
3) Get at least 50% of the maximum Cs while doing so.

Champions that can pull this off:
Alistar
Cho'gath
Galio
Garen
Gragas
Morgana
Renekton
Soraka
Swain
Yorick

Champions that come close:
Anivia
Blitzcrank
Cassiopeia
Heimerdinger
Irelia
Maokai
Nunu
Singed
Skarner
Tryndamere
Urgot
Vladimir
Volibear

Generally the 'close' champions are just missing one key part, often the sustain. Ie, Anivia with lots of potions can be viable for sustaining through the harass, while naturally having the ability to farm well and punish a tower dive (stun, Q, wall).

So the big question is where do you put your AD carries? If they jungle, they aren't really able to provide much in the way of ganks (only Ashe really has real CC, Vayne has a knockback with stun on good positioning). So maybe run them in the top jungle and hope top lane can handle life without many ganks.
If you put the AD carry mid, then you have to put an AP carry top or bottom, since they are too mana dependent to reliably jungle and still gank at all.
Potentially, you can run Swain or Morgana bottom, with your AD carry mid, bruiser/tank top, and the proposed junglers, but a 2 champion pool becomes awfully easy to ban around. Even adding AP Galio or Gragas to the mix only helps a bit.

No offense, but it feels like a good deal of the PBE is extremely incompetent. I'm not even saying I'm a good player, but anyone I've played with/against have just been head over heels retarded.
That being said, there isn't any real way to actually test this on the PBE server.

Either way, double jungle sounds terrible. As it is now, jungling is still extremely slow. The banking system helps it out a little bit, but if you're constantly killing them with two junglers, they're not going to be banking very much.

Invading the enemy team is usually pretty stupid as it is now as a single jungler. If you're playing against players who actually understand how to ward, you'll never be able to cross the river without the other team knowing. Even if you bring flash and exhaust, they will each only help you once every 3 or 4 minutes, which doesn't give you many options while they're down.

In terms of laning, the bottom lane 2 could very easily deny the 1. The number of people in a lane does not constitute faster pushing if you're playing against people who actually understand the concept of last-hitting. As long as they don't push the lane up, they can easily deny the 1. Either that, or they could just very easily push the tower down extremely fast, while the gap between exp is still too little for the 1 to be able to stop them, and both junglers are busy gimping themselves while they wait for the small camps to respawn on their side of the jungle. Not many junglers can actually gank at level 2. (Nocturne, Rammus, and Fiddles, all pretty half-assed. Pretty much only Lee Sin.).

Lastly, your proposal does not actually leave much room for a decent team comp. If only tanky characters are allowed into the top and bot lanes, that leaves AP, ranged AD, and support to be jungling and mid. No matter how you arrange that, one of your most important damage dealers are taking smite over something that would suit them better, or your support is bringing heal/cv/smite without any room for escape.

Right now, the PBE isn't that far from Live in balance. If you actually want to see this happen, find a friend to duo queue with. Better yet, find a 5's team, get high elo, and make it the "new meta". See how far you can get in ranked, and prove me wrong.

Have to try this out sometime!

I used too do this with a friend way back when people didnt always have a jungler of there own in games. It worked great. But only if there was no enemy jungler. Now with the creeps spawning so fast it could work again.