I'm currently doing my law masters abroad (in UK) and so far I am really enjoying research and discussion, which is so different then undergrad level. When I spoke with my advisor he said it is possible to do a PhD in our school, which would save some time since it is a familiar place and they already have all of my documents. However he said Russell Group Universities (which is supposed to be Ivy League in UK?) do not give any teaching opportunities to people without an actual PhD. So hearing that I will not interact with any students and just research on my own and pay a lot of money without earning any real job experience made me question my decision. My advisor openly said it is unwise to do PhD without any teaching and I should do it part time and look for jobs in universities in the same city (and I looked - there are none) or get professional qualification to have some backup plan. The problem is I am from a civil law country and qualifying as a solicitor here would take extra 4 years, it is not even a logical plan to spend 4 years on my back-up option. I really want to stay in this city and continue in this established university but I don't know what to do, what if I am not cut out to be a lecturer in the future?

The Russell Group are the group of the most research intensive universities so a bit different to the Ivy League. Your advisor is correct to say that they are unlikely to employ anyone as a law lecturer without either (and as I understand it preferably both) a PhD and substantial professional legal experience. I would also doubt that you'd have much luck getting a lecturing position at a lower-ranked university without the ability to teach English or Scottish (depending on where you are) law, which you presumably don't have, as the majority of an LLB is the compulsory qualifying law modules and only EU law (if you have studied that) would really cross over to a civil law training. I suppose you could do the 1 year graduate diploma in law to convert to English / Scottish law, but I suspect those with the qualifying degree plus either PhD or legal experience would still be preferred for any lectureships.
As to doing a PhD, you might be able to get some hourly paid teaching for seminars - certainly I knew people who did a little of this during their law PhDs at the Russell Group university where I studied. But a PhD is a research degree and it is a lot of lonely research, and if that does not appeal, don't do one. Not least because even if you were one of the minority of people, who get a lectureship afterwards (most do not - and those who do rarely get them at the university they qualified at), then more lonely research is a large part of the job.

Quite unlike the USA and Canada where you routinely take up Teaching Assistantships (TAs) which enable you to acquire undergraduate teaching experiences, this is not a given in UK universities. You could obtain a UK PhD without teaching for a minute and without publishing a single paper! Your PhD is solely based on a piece of research documented in a thesis format. Without the requisite teaching experience, you would be EXTREMELY LUCKY to get a lectureship in the UK. In the USA and Canada, you'll be completely weeded out by the search committe. This issue really makes me think that whilst the UK PhD equips and prepares you to becomne an independent researcher, it doesn't prepare you to become an academic (i.e. a university lecturer and researcher). If I knew this when starting my PhD, I certainly would have headed for North America (USA or Canada).

If, you can't find a funded PhD which comes with TA in the UK and your ultimate goal is to Lecturer, forget it. I am talking from personal experience: with a first class honours degree, a masters degree with a distinction and a PhD -- all from Russell Group Unis -- I still couldn't secure a Lectureship close to 2 years after finishing my PhD, ostensibly because I have no previous university teaching experience. The RA job I got is nothing but 21st century slavery. I thought the department would allow me co-teach undergraduate modules, gratis and pro bono, so I could put it on my CV. No! They cared less. I simply gave it up, picked up my teacher certification and currently teaching in a high school. Not precisely what I hoped for after gaining my PhD.

I did my Masters at a Russell Group University and I have a number of friends from my Masters who are doing their PhD there who regularly teach. It may be dependent on the course/department, but I can think of at least 4 or 5 people who I know who teach as part of their PhD. They are restricted as to how many hours they are allowed to do. But it is definitely not the case that Russell Group universities do not allow PhD students to teach. It may be worth enquiring with the university/department that it is you are interested in undertaking the PhD with.

I've recently started a PhD at another institution (not Russell Group!) and whilst my contract states that I am allowed to teach up to 6 hours a week as a full-time PhD student, my supervisor is extremely resistant. I can understand that at this point I am very early in my research, but if he still is against it in my 2nd or 3rd year then I will be extremely disappointed and frustrated as -- just as others have mentioned -- it is hard to secure an academic position thereafter.