If I had to be left with only one collection of American folk songs, it would be "Folk Songs of North America". It really is a gem. As mentioned above, it has songs from other collectors. Alan Lomax obviously wished to be very inclusive and to give a good broad sample of English-language North American folk song when he put it together. My 1960 edition has a map of North American folk song styles on the inside cover and a basic discussion of guitar and banjo accompaniment; they're very nice pluses. Because we're on the topic of songbooks, it's good to note that the latest edition is certain to have a good current bibliography.
For general British Isles songbooks, I like "folksongs of Britain and Ireland" by Peter Kennedy, and "Songs of England, Ireland & Scotland" by Milner & Kaplan. These are the general collections recommended at the English Folk Dance and Song Society library and both are still in print. Again, good bibliographies.
If you want Irish Songs, the best choices, I think, are... the O Lochlainn collections, "Irish Street Ballads", vols. 1 & 2 (now printed in one book) and Sam Henry's, "Songs of the People" which is a collection of some 690 songs printed in a newspaper column in Northern Ireland in the 1920's and 30's.
There are many different types of song collections. "Rise Up Singing" is obviously a quick reference book. Others are scholarly looks at regions or professions, or local community resource books, or archival collections, etc. Those that aren't great are certainly good. God bless all the people who kept a snapshot of these "songs in flight" for us!