Usually, fewer people apply in 1st Round, but they tend to be the better prepared candidates - so see for yourself what type of competition you would prefer.
Oxford adcom state that the chances are equal in each round. In my opinion, it's not exactly so: if you come from overrepresented demographics, it's better for you to apply there as early as possible (but it's true for every school, I think). And, vice versa, if you have a truly unique profile or there are only a few applicants from your home country, you might have even better chances in later rounds.
For Cambridge, I'd say it's hard to predict which time is better. Last year, 1st round was very hard, 1.5 - almost impossible, but 2nd and 3rd seemed to be easier: they invited and admitted quite a few people from their pre-interview waitlist.
Cranfield SOM has rolling admissions, and they advise to apply as early as you're ready. It makes sense: as soon as they have a full class, they just stop accepting applications, and it could happen very early in the season (we have a thread about this on the forum).
Can't say anything about LBS or other UK schools - perhaps, other clubbers could help you here.