All things being equal, would a person have a higher chance of getting admitted to study towards a PhD at the top 3 UK institutions (Oxford, Cambridge, LSE) if they stress they want to become an academic or if they have a conditional offer to start working for a prestigious institution upon completion of their PhD (ie.The Office of the Legal Advisor of the Ministry of Foreign Affairs)?

In the event of the latter, for example, is there a greater chance of this applicant being viewed as a "unique candidate" and a "positive statistic" upon graduation? Or (in the event of the former) is the prospect of going into academia and thus being likely to add on to the number of academic publications associated with the schools name of greater significance?

I'm just not sure they'd care that much as between the two. I think they're more concerned that you have a reason for pursuing the PhD in the first place - it is a long and involved process and I suspect they just want to be sure you're fully committed to it. They'll be far more concerned with whether you're a quality candidate right now than whether you're going to pursue one of two equally laudable careers when you finish.

Moreover, if you can't explain to yourself why the PhD fits with the goal you have for yourself on completion, then I think you have a more fundamental problem than tailoring your application to fit their perceived preferences.

I'd agree with what Canadian1 said. I think three questions will be on the mind of the admissions tutor:-
1. is this person academically bright enough;
2. does the proposal make sense' and
3. can we support the PhD - is there a member of staff who has the knowledge and time to supervise and do we have sufficient library materials (though staff interests and library materials often fall in the same areas!)