If the death or obituary was worthy of national attention, the Times Digital Archive is worth a look.
The index is available here and subscriptions can be taken out at this site. Many libraries and other institutions already have subscriptions to the archive, so it is worth checking to see if you can access it locally free of charge.
There are also online collections of local newspapers, the most significant one being the British Library’s 19th Century Newspaper Collection, accessible at the British Library and via many other libraries and institutions.
Once again, you will need to find somewhere that has access to the database or take out your own subscription. Have a look at http://newspapers.bl.uk/blcs for more. This collection is growing all the time, and is worth returning to every time you want to search for an obituary.
There are other, smaller, collections of online resources, so it is a good idea to search on the internet for the location you want to see if you can take advantage of such a shortcut.