My understanding (from a non empirical source, ie DS) is that LSE’s international reputation is on the rise. There is always a level of subjectivity, but it would not be unreasonable for an LSE professor to consider the LSE to be the top UK department. (Obviously you might get a different answer if you asked a Cambridge professor.)
The important thing to realise is that the LSE, UCL, Cambridge and Warwick degrees are very mathematical. This does not mean they are “better” than degrees from, say, Oxford or Durham. However the course content is different and to some will be quite dry.
In terms of University experience it suited my son. He wanted to live and breathe economics and found plenty of like minded peers from different backgrounds in the UK and from all over the world. He apparently only went to one freshers event and didn’t enjoy it much but was active in several societies, played in a 7 a side football league and regularly joined others to eat in China Town or groups, including international students, would meet up and cook together. The Athletics Union was known to be quite raucous, with pre-pre-pres starting at 9.30 am on their big night. But they were the exception rather than the rule. As with most Universities if you try hard enough you should find your people.
Another thing to consider is that the degree is rigorous. If you don’t work hard you could end up failing. Overseas students pay high fees and normally work hard. But so do the successful British students.
There is some family debate as I also read economics at LSE whilst DH read PPE at Oxford. (DD rebelled and opted for medicine.) DH is the most literate but even back in the day avoided LSE for his Masters because of the maths requirements. Breadth or depth. Some employers, especially those looking for economic forecasters will prefer depth and the quantitative skills LSE teaches. And because the department is so large there is huge scope to take interesting specialist courses in your final year. (DS ended up graduating in Econometrics and Mathematical Economics, which is an area where LSE can realistically claim to offer Europe’s strongest UG degree.)