Quick answer
The United Kingdom does not use one single property registry process for every property. England and Wales, Scotland and Northern Ireland each have their own registry system, document names and search routes.
England and Wales
HM Land Registry covers England and Wales. GOV.UK explains that users can search property information and obtain title register and title plan information for registered property in England and Wales.
Official source: https://www.gov.uk/search-property-information-land-registry
GOV.UK also explains that a title register can usually include ownership details, title number, last sale price where available, mortgage information and rights or restrictions. A title plan shows the property's location and general boundaries.
Official source: https://www.gov.uk/get-information-about-property-and-land/search-the-register
Scotland
Registers of Scotland is responsible for Scottish land and property registers. ScotLIS guidance explains that Scottish property documents can include a title sheet and title plan, and that the title sheet may include property description, registered owners, securities or charges and burdens or responsibilities.
Official source: https://scotlis.ros.gov.uk/help-and-guidance
Registers of Scotland also explains that the Land Register of Scotland records ownership of land and property in Scotland and can be searched through ScotLIS.
Official source: https://www.ros.gov.uk/scottish-land-register
Northern Ireland
Land & Property Services keeps the Land Registry for Northern Ireland. nidirect explains that Land Registry is a record of registered land, houses and properties in Northern Ireland, and that if land is registered you can see folio details including current registered owner and rights or charges affecting the folio.
Official source: https://www.nidirect.gov.uk/articles/searching-land-registry
Why this matters when ordering
The right document name and expected contents can change depending on the property's region. A Title Register and Title Plan are familiar England and Wales terms. Scottish searches may use title sheet and title plan wording. Northern Ireland searches commonly refer to folios and property maps.
Independent service note
LandRegistries.com is independent and is not HM Land Registry, GOV.UK, Registers of Scotland, Land & Property Services Northern Ireland or any government department.