I've never heard of those 18-19 London salaries. Woah! Finance though surely?
Though you'll get to higher tax brackets with that pretty quickly too, so will likely want to leave a fair share of revenue in the company (you still need to pay 20% corp tax on profit), or put it into your pension (tax relief on up to £40k/y).
If you pay yourself a salary from a ltd company there's ~14% Employers NI, 12%-2% Employee NI Depending on tax band, 20-40% Income tax depending on tax band. If you pay a dividend there's 19% Corporation tax plus 7.5%-32.5% Dividend tax.
So 46%-56% tax on salary and 26.5%-51.5% tax on dividends. Nowhere near 13%, unless you're talking about 'creative accounting' by running non-business expenses through the company to effectively get them tax free.
I'll have to enquire with the few mates there -- don't see a reason as to why they'd bullshit me. Frankly I have nfi however, apologies for the cop-out of an answer.
Sole traders are different to one-person Ltd companies. As a one-person Ltd, you used to get a lot of tax breaks, but over the last 5-6 years they seem to have been steadily eroded.
Anyone paying a rate as low as 13% now is probably using a borderline-legal grey-area scheme, which is dangerous because HMRC not only hit those folks up for back taxes when they find them, but put fines on top too.
I don't live in UK anymore and I could do something similar in my location, but I want to pay taxes and my effective rate is good.