The Mitchell Library in Glasgow (at Charing Cross) holds a collection of local and national newspapers. Details of their collections are online and this includes access to ProQuest Newsstand - a service similar to Nexis - as well as historical back runs of The Herald and The Scotsman.
National Library of Scotland’s (NLS) newspapers collections site has information about the library’s printed and electronic newspapers as well as lists of newspaper indexes. Some of the electronic materials may be available if you register with the NLS, or if you visit one of their sites: Kelvin Hall Glasgow, or George 4th Bridge in Edinburgh.
Library Hub Discover allows you to search the collections of over 100 academic, national, and specialist library catalogues across the UK and Ireland to find out where a specific newspaper or journal is available to read.
In London, the British Library has an extensive newspaper collection which can be consulted in their reading rooms, it covers all UK and Irish newspapers plus nearly 100 international titles.