Why do North American law schools require an undergraduate degree, when the UK's don't? I'm aware that most NA law schools require an undergraduate degree. For example, Yale's application website states:


  You must receive, or expect to receive by the summer of 2020, a bachelor's degree (or the equivalent) from an approved undergraduate institution in order to be eligible to apply. All offers of admission are contingent upon graduation.


In Canada, McGill doesn't require a degree, but strongly recommends it for competitiveness (link ):


  While candidates who have completed 60 credits of university study are eligible to apply to the Faculty of Law, admission to the program is competitive and as such, almost all students admitted in the “University” category (see below) have completed an undergraduate degree.


That said, a law degree is an undergraduate degree in the UK. According to this website:


  Students have the choice of studying a qualifying law degree at a wide range of UK Universities immediately after high school or after they have completed an undergraduate degree.


Thus a UK barrister or solicitor will spend between 4 and 5 years in university (4 = 3 yr qualifying law degree + 1 yr for LPC or BPTC, 5 = ≥3 yrs first non-law degree + 1 yr GDL + 1 yr LPC or BPTC), while the average North American lawyer will spend closer to 7 years (4 yrs undergraduate + 4 yrs JD).

Many senior UK judges have just 1 3-year undergraduate careers, some not in law at all. For example, Lord Sumption has just a BA in History from Oxford, Lord Phillips, first UKSC President, has just a BA in law from Cambridge, and Lord Neuberger has just a BA (yes, not BSc) in Chemistry from Oxford.

The UK's leading law schools look better than Canada's. The 2018 Times Higher Education World University Rankings ranks Cambridge at 5, Oxford 6, UCL 8; but the topmost Canadian are Toronto (10),  McGill (13). The QS World University Rankings by Subject 2017 ranks Oxford (2), Cambridge (3), LSE (7), UCL (12); but the topmost Canadian LS is Toronto (17). 

It doesn't appear that the far briefer university education has affected the quality of UK law or lawyers. 

To that end, I'd reckon that to save students time and money, North American law schools ought follow the UK's lead in not requiring UGs.

Note that to simplify this question, I haven't discussed other similar nation-states  like Hong Kong, or asked the analogous question for medical school.
Note that a US undergraduate degree is very different from a UK degree. The latter is much more specialized. In UK it is assumed that the general education occurs before the university level. 

But in the US, the degree is quite general. Students of, say pre-law also study things like history, philosophy, foreign language, etc. It isn't that different from a History major. The "major subject" in the US accounts for half (more or less) of the total credits required. Even a math major will study all those things in the US. 

And, in both places, it is generally believed that a lawyer needs that broad grounding. It is really just a question of where and how it is obtained.