Hey, bit of a late reply but thought I'd put down my thoughts on the off chance you haven't made a decision or for the benefit of anyone else reading this thread.
Well done on getting offers from three great unis. If all goes well you'll get a great degree, and will be set up well whether you decide to stay in engineering or chase the money. You're right, Imperial has the best rep for engineering, probably followed by Warwick imo (I went to Warwick, so I'm biased!)
I was in a similar position to you when I applied (a few more years ago than I care to remember) - I had the interest, but wasn't sure what to specialise in or whether I wanted to do it as a career.
I chose to go to Warwick as (as you note) the structure of the course allows you to put off choosing your specialisation until the third year. This suited me down to the ground as I had no idea what I wanted to do at 17. There was (and still is, I guess) quite a lot of business content in the first two years of the course at warwick (may not be the case at the other), so you'll get a feel for whether you enjoy the business side of it vs the "real" engineering. Note that this will be taught by WMG (not to be confused with WBS). I found the quality of the WMG lectures really good, although it may be more engineering specific (lean manufacturing, supply chain, for example) than what you get at LSE.
If I were you I'd have a goodd think about whether you want to spend the next 3/4 years of your life in London, or in a campus in the West Midlands. Your lifestyle will probably be incredibly different (not necessarily better/worse) at Warwick compared to London so that would be the aspect I would focus on for this decision.
Good luck!