I don't think there will be much of a meaningful distinction between the two just based on what you actually learn or the branding.
The Kings MSc would give you a longer run way if you're using this to have a more favourable visa transition (from Student to Work or equivalent) (but also push out your journey as a lawyer and it would cost more) to apply for various cycles. It also seems substantively geared to being useful, although whether it is actually meaningful when compared to what you can learn by yourself (if you apply yourself in the right way) or what you'd learn on the SQE, I could not say. However, I know that as someone in practice, that what you'll learn is unlikely to be materially helpful.
UCL LLM will have a tighter runway (with respect to visa transitioning), which means it could be pretty intense (e.g. juggling apps, social life, studying, etc). UCL edges in branding but it's not really meaningful. Although more people would be familiar with the UCL LLM than the Kings MSc which appears quite new.
In theory you could learn the SQE material by yourself without doing either courses. You could also learn the SQE material alongside your degree, if you think you'll need more time to prepare for the SQE, you may be minded to take the option which gives you the most time. However, going straight into a law firm with just an SQE and a MSc / LLM is a relatively unusual path, as you would qualify for associate roles but you would not have any practical work experience bar your experience you have had as a foreign lawyer.
I would flag that you could always try for paralegal roles (if you have a visa) and apply to vacation schemes / training contracts in parallel. I have seen quite a few people find success with that, compared to those that I have seen going into further education and applying for the same opportunities.
