Lycra is used in such small amounts that it's not necessary to take it into account when dyeing. What most people call lycra is actually a knit nylon with lycra knitted in which as you cay is easy to dye. Most of the time
Part of what affects ease of dyeing has to do with weave.
http://www.pburch.net/dyeing/FAQ/spandex.shtml (I think written before the iDye stuff
http://www.pburch.net/dyeing/dispersedye.shtml )
I only ever dye wool with cold water dyes (in a large 60L plus container and outside) I usually use warm water to bath temperature as a lot of the fulling process (the shrinkage and matting) occurs with aggitation and shock of going from hot to cold. A lot, not all! Boiling definitely shrinks woollens like mad and again the weave has a lot to do with it (crepe will shrink by up to 30% in one dye bath it seems flannels take a little more work). In thepry a worsted should hold up better as it is spun with all the wool fibres in the same direction (scaled of the hair lie in one direction) while in woollens they are in both directions (so the scales will catch and grab each other).
Anyway, long story short: both will dye pretty with Dylon cold water dye, for example, well but be careful to wash the fabrics first and make sure to really work the dye evenly as cotton is prone to uneven blotching (it absorbs dye very quickly) and so can wool be (possibly natural oils or the physical nature of the fibre).