Okay, I'm not sure if this was really asked for or not, but here goes:

In general, to practice law in the UK (as well as many other countries), you just need a bachelors degree, usually an LLB. This is often followed by a couple years of job training. Alternatively (at least in the UK), if you didn't get your bachelors in law, you can take a year-long conversion course which leaves you in approximately the same location. This is important because it's a lot harder to switch your subject in UK universities than it is in US ones. At least at Oxford, past the first term of the first year, many people who wanted to switch subjects were forced to either wait until the next year or even in some situations to reapply to Oxford altogether.

The situation is a little more complicated in the US as each state has its own requirements to take the bar exam. Most states won't let you take the bar unless you've been graduated from an accredited law school. Some states will allow only JDs to take it. Some states will allow both JDs and LLMs to take it. Only a few states (California being the main one) will allow you to take the bar from a nonaccredited law school.

There are quite a few different types of LLMs that serve various purposes, but the generic LLM is usually a one year program for those who have gotten a law degree in their home country and want to practice law in the US. (Other LLMs are sort of like masters degrees for people with JDs. Tax is the main one, but there are others). LLMs have a hard time finding jobs within the US compared to JDs (see below). Many of my LLM friends have had offers from US firms to go back to their home country and work at their office branch there. I think a law firm would be pretty puzzled by an American with a generic LLM instead of a JD.

JDs are (usually) a three year degree on top of a bachelor's degree. You do NOT have to have any law background to go to law school for a JD. In fact, if you are going to undergrad in the US, I'd highly discourage you from trying to do so. At best, it will not help your application. At worst, it will actually detract it. For what it's worth, lawyers with technical, scientific knowledge are in high demand right now, so if you're scientifically minded, getting your degree in a science and then going to law school is an option.

Law firms, like many places, are going through some pretty radical changes right now. It's hard to tell where it will end up. That said, law is one of the more elitist professions. "Top" law firms care very deeply about which law school you attend. To pick two Californian examples, Boalt (law school name for Berkeley) will give most of its students good jobs when they graduate. Those who come from La Verne will have a decidedly rougher time. Since you're potentially interested in prosecution, the DAs office cares about where you go to school, but not quite as much as the top law firms.

One of the incorrect things I've seen described here is that you "should go to a local school because that's where the bar is." This is true of many mid to low ranked law schools who teach with the bar in mind a great deal of the time. However, the "top" law schools do this far less. You learn how to pass the bar in a two month long prep course called Barbri. That said, if you're not going to a top law school that has national or at least regional reputation, go for the local school. You'll make better contacts, and networking is of huge importance coming from a lower law school.

One thing to keep in mind is just how damned expensive law school is. People regularly wrack up 100k+ debts at law school. And law school, especially in these uncertain times, is no gold ticket to being able to pay off those loans. Even if you're able to get one of the top jobs, many end up feeling trapped by terrible jobs because they want to pay off their loans and other jobs pay too little. Many would laugh at the idea that the 60k starting salary most California DAs pay makes it hard for ends to meet, but when you consider 4 years of a BA debt plus 3 years of 40k+/year law school debt plus cost of living being what it is in CA, for many it is not an easy life. Also keep in mind that, unlike undergrad, there really is no such thing as needs based aid in law school.

A last piece of advice: take your time once you get there, to evaluate whether you want to go to law school. Do not go straight from undergrad to law school. Even taking a year off is worth it.
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I've tried to address the main issues I saw from this. I know it's not directly to the point of "US vs. UK" but I thought it might be helpful in making your decision.

If you're considering either Cambridge of Oxford, I can give you more detailed insights. But if not, I won't confuse things by providing Oxbridge specific information.