In order to settle in the UK purely on the basis of wealth, you need to apply under the Tier 1 Investor route. Under this route, the applicant is required to invest £2,000,000. They will gain settlement after 5 years. This can be decreased to 3 years by investing £5,000,000 or 2 years by investing £10,000,000. At application time, they must prove they have £2,000,000 available in a UK bank account, and if they've held the funds for less than 2 years, they must prove its provenance. When extending the visa or applying for Indefinite Leave to Remain, the applicant needs to show they've invested the required amounts in actively trading UK registered companies for the time covered by the previous leave. They cannot (anymore) just invest the money in gilts, but by investing in a large number of well established UK companies, there is practically zero risk involved. The route has been criticised as being an excellent vehicle for money laundering, although some reform has been introduced requiring applicants to complete criminal background checks.
The rules for this route are fairly complex, and very few applications are made annually. It would be advisable to make this application with the help of an experienced (and expensive) solicitor. But anyone in a position to apply as a Tier 1 Investor obviously won't have any problem affording this.
Under this route, the applicant is allowed to work, except as a professional sportsperson or coach, and there are some restrictions on working as a doctor or dentist. And they can also bring over their dependents to the UK.
Taxes are dealt with in the same way as any other UK tax resident. With the amount of investment required, and the high income that they presumably have, they would likely be required to complete self-assessment, but they will probably pay an accountant to do it.