There are plenty of international students who are lawyers in the UK, both solicitors and barristers. You just need to be realistic as to your prospects of succeeding in what is a very competitive industry in this country.
Just on the ULaw point, even if you were talking about an institution that does have a strong reputation for its students obtaining training contracts or pupillages, the key point to remember is that those candidates will have had their success because they are very good candidates, not because they went to a particular university. Going to a good university can give you advantages because of the superior quality teaching or opportunities that they provide, but the best universities also recruit the best students out of school, and that is generally the main reason behind students at better universities having more success. It's because they were better candidates to begin with, not because of some magic influence that their university has. That is not determinative, of course. I know some very good barristers who went to 'bad' universities, and I have seen many bad candidates for pupillage interview who went to Oxford, Cambridge and Russell Group Universities. Better universities having students that have more success is just a trend and doesn't guarantee anything, but it is about the quality of candidates ultimately, and not their university.
On the criminal law front, you have been told wrong. Criminal law is, relatively speaking, more accessible than other areas (though still competitive), but it does not pay well and it can be much more difficult for junior practitioners to get by from a financial perspective. By contrast, commercial law and intellectual property law are both very competitive areas to secure training contracts or pupillages in.
You've been given a lot of information about the solicitor route. On the Bar side of things, you need to do a qualifying law degree (either an undergraduate degree, GDL, or possibly an LLM that acts as a qualifying law degree), then the Bar course and then you need to secure and complete pupillage. Your bar exams are taken on the bar course; they are not a separate thing. Statistically securing pupillage is even harder than securing a training contract, so you really do need to be sure that you have a realistic prospect of obtaining a pupillage before you commit to the time and effort needed to go down that route.