The degree can be converted. So a common law LLB degree makes you eligible to qualify in a range of common law countries. So you can go and sit the bar exam in the USA if you wished. What limits you is if you qualify in the UK as a solicitor or barrister, then you cannot just go and practice in another country. You'd have to qualify again in the new jurisdiction.
& don't listen to Ser Alex Toyne. Contacts are always going to be useful but you get out what you put in. If you do well and take the time to seek vacation schemes/mini-pupillages and are friendly with the staff at university then you never know what will spring up. The law students who complain that contacts are necessary are the ones who either didn't do much during their degree and couldn't land a training contract last minute or they want to be chancery barristers or something when yes contacts become important.