If you do not have a physical base (office or branch) there is no requirement for overseas companies to register in the UK but if you do have a physical presence, then you must register your company with Companies House. Partnerships, LLPs, unincorporated bodies and government agencies cannot register as a limited company in the UK.
Overseas companies must register with Companies House (or Companies House in Edinburgh for companies registered in Scotland) within one month of opening a UK establishment. They can do so by filing Form OS IN01 'Registration of an overseas company opening a UK establishment' with Companies House - along with a £20 registration fee. If a company is registering its first UK establishment, it must additionally provide:
a certified copy of the company's constitution documents (including a certified English translation if required)
the latest set of company accounts (including a certified English translation if required) if it is required to file these in its own country
if the company is required to prepare, audit and disclose accounts under parent law, then they need to provide a copy of the latest set of accounts that had been disclosed as of the opening date of the UK establishment (including a certified English translation)
If you’re looking to set up a business in Scotland, you can get help from Scottish Development International (SDI).
For more information, read Registering an overseas company in the UK.