I should have spotted this one before now. The first UK national newspaper to make its archive publicly available (as opposed to The Times, which is only available to institutional subscribers) was not The Guardian, but The Scotsman. At http://archive.scotsman.com you can access every issue of the paper 1817-1950. Searching is free and gives you the headlines. To view the full article you need to take out a subscription: a 24 hour pass is £7.95, a 48-hour pass £12.95, a weekly pass £19.95, and so on. Not cheap, but the search system is a good one, and the results promising: over 1,000 hits for the word ‘cinematograph’ between 1895 and 1930, for example.
I’ll do a round up soon of all these online newspaper archives of relevance to silent film research.