As IanC pointed out, people like Carthy and Jones have worked mainly from traditional sets of songs recorded from traditional singers by the many people who collected in, principally, the late 19th and early 20th centuries.  Collection still continues, though obviously the scope is more limited nowadays.  Broadsides are sometimes used as sources (more often to bulk out an incomplete traditional text than on their own), but as has been pointed out, they tend to come without a tune direction, at least so far as 19th century examples are concerned.  Earlier song publishers like Chapell, Motherwell, Ritson and others included texts and melodies found in tradition with contemporary compositions, and of course Burns wrote many of his songs to traditional tunes and devoted a lot of time to collecting them.

The subject is too complicated to deal with briefly; Dave Harker's book Fakesong (Open University Press, 1985) provides a detailed history and analysis of the major collectors and publishers from 1700 onwards.

A lot of the work is still to do, so you needn't expect to have it neatly packaged and available for you on the web, though Bruce Olson's site which Kat linked to above has a great deal of information available; he deals mainly with pre-19th century broadside material.  Tune indications were more common then, though often they referred to titles which are wrong or unknown today.  Bruce's site is by far the most useful privately-maintained one, and is the product of many years' research: be warned, it requires patience and thought to find your way around it.  There are also links to other sites which you will find helpful.

Simpson's book is no longer in print, but you can get it through secondhand dealers, usually not for less than £70 or so.  Child avoided using broadsides in the main, as he specifically wanted material taken from tradition; he also included only a few tunes, by way of appendix.  Bronson produced a huge compendium of traditional tunes for the "Child" ballads, but this too is out of print and typically goes at around £170 per volume (there are 5).  Child, likewise, is unlikely to be found cheaply; you can expect to pay no less than £200 for the 1960's paperback reprint, though patience and persistence can sometimes find it cheaper.

For a useful -though inevitably incomplete- list of some of the more important British and Irish collections of traditional song that are still in print, see  David Herron's Chapbook  at the  South Riding Folk Network  site.

Malcolm