Richard Heaton provides a very useful Index of British and Irish Online Digitalised Newspapers - lists of all the titles of British and Irish Newspapers (31 different collections covering 17th to 21st Century) now available online, providing links, subscription details and dates covered.
Their reading room in Boston Spa provides free access to "the only large, integrated, national newspaper service in the world, combining facilities for the collection, preservation, and use of newspapers all on one site".
The Library provides, in collaboration with FindMyPast, online (subscription) access, to their newspaper collection, through the British Newspaper Archive service. (As of July 2016 this had 656 titles online - see their coverage and video guide.)
The British Library's British Newspapers 1800-1900 - "managed by Gale Cengage, contains 49 national and local titles with two million pages of searchable digitised content". (Accessible to subscribing libraries and UK further and higher education institutions.)
Newsplan - a co-operative programme (involving between public libraries, the British Library, the National Library of Scotland, the National Library of Wales, the National Library of Ireland, and the newspaper industry) for the microfilming and preservation of local newspapers and for making them accessible to users; links are provided to regional Newsplan projects, a number of which have already produced extensive online catalogues.
The London, Edinburgh, and Belfast Gazettes are the official newspapers of record in the United Kingdom. As well as the current editions, the website includes searchable archives of the complete gazettes from WW1 and WW2 containing medal awards, army and navy commissions, promotions and much more.
Historical Newspapers - a (subscription) service incorporating The Official Index to The Times, Palmer's Index to the Times, The Historical Index to The New York Times, and Palmer's Full Text Online, 1785-1870.
The Nineteenth-Century Serials Edition (ncse) is a free, online edition of six nineteenth-century periodicals and newspapers: The Monthly Repository and Unitarian Chronicle, The Northern Star, The Leader, The English Woman's Journal, The Tomahawk, and The Publishers' Circular.
Internet Library of Early Journals - a project that digitised 20 years or more of The Annual Register, Blackwood's Edinburgh Magazine, The Builder, The Gentlemen's Magazine, Notes and Queries, Philosophical Transactions of the Royal Society (free).
Gales News Vault - provides simultaneous searching "across The Times Digital Archive, 17th and 18th Century Burney Collection, Financial Times Historical Archive, The Economist Historical Archive, and many more" (subscribing libraries)