There are no legal requirements in the UK to set up as a nail technician. You can quite literally wake up one morning and decide to start working on clients. However, you can't get insured without an accredited qualification, and you're unlikely to get (and more importantly retain) clients unless you know your stuff.
Clients quickly pick up on poor skills and vote with their feet ie.go elsewhere.
I whole heartedly believe you cannot learn a brand new practical skill via on-line learning, especially something like pedicure, nail extensions, etc. They are practical skills that can be tricky, only actually being shown live, or corrected there and then whilst you train can be mastered.
On-line is fine for skill building, enhancing what is already learned and things like nail art, but basic skill are best in person.
I didn't go to college to do my training, I did it via private individual training academies. I've done various skill builds with various companies and brands, even 1-2-1's with private tutors.
There are plenty of training options but do your research