This is an interesting dilemma, since you have a newer 1-yr program within a very established, high-ranking business school (LBS) vs. an established 1-yr program from a slightly lower ranking business school (Judge) that is also part of one of the world's most famous universities (Cambridge).

Frankly, one could make great points on either side of the argument above, if only looking at those basic facts.  What you need to do, however, is figure out which of these two options is right for YOU.  A lot of that depends on your career plan and your fit with each program. 

On the career side, the tracks for consulting, technology, and finance (VC) are all extremely different, so it would behoove you to first really narrow your focus when it comes to post-MBA targets.  It will be incredibly hard to attempt to recruit into three possible fields while attending a high-intensity, short-duration MBA.  A backup plan is great to have, but you need a single primary plan heading into b-school; this will shape course selection, outside activities and much more. 

For the sake of argument, let's imagine you've picked consulting and will be looking at MBB, Kearney, etc.  You'd then need to do your due diligence and talk with the consulting clubs, career services teams - all while combing through the career reports to get a feel for which program might be better suited to help you reach your short-term, post-MBA goals.

On the fit side, I'm not sure where you are located, but you might do well to spend some time at each program/on campus to get a feel for the programs.  As you may know, with LBS, you will be mixed in with the 15 and 21 month folks for some courses and activities - tapping into a broader network.  And at Judge, you will be assigned to a college within Cambridge, which comes with an additional network as well.

Hopefully these steps help you begin to think through this choice.  Best of luck and thanks for your post!