If you're not required to use non-toxic loads, I think good quality lead shot target loads are fine choice. Contrary to popular opinion and marketing, there is no practical advantage to very high speed lead shot loads for efficiently killing small birds in the uplands. There is always a trade off with more velocity and heavy payloads, and the target shooting community has figured out what works best with repeatable results. The main difference between good target loads and most hunting loads is the quality of the shot. Best quality target loads are almost always a better choice for hunting small birds because they use the best quality ( hard) lead shot and produce the most uniform patterns.
The currently fashionable cheap hunting loads loaded with soft shot at high velocity are the worst choice - pattern poorly, recoil more. Soft shot loaded to high velocity usually patterns poorly because of deformed shot and "blown" holes in the patterns. Cheap target loads are also loaded with cheap, soft shot, but at least most are at a sensible velocity so still pattern reasonably OK and don't beat up the shooter with excessive recoil.
If you really want the best, choose something like ( for example) Federal Gold medal, Winchester AA or Remington STS, loaded at no more than 1250 fps. You will bring home the birds.
Snipe are definitely defined as migratory birds, and hunting them requires the use of non-toxic shot Canada wide. You'd have to check your regs in your province about Woodcock, but I suspect they are also considered migratory and Federally regulated.
In comparison, steel hunting loads actually benefit from speed. Steel pellets don't deform under firing pressure and their light weight can be somewhat compensated for with more velocity. But it's hard to make general recommendations for steel upland loads. You will need to pattern your gun to find out what works best for you with steel.