Advance booking discounts are offered for some (but by no means all) routes, directly by the train operating companies. The sites you refer to simply act as agencies and don't offer any fares that you can't get directly from the train operators. (The only 'extra' you get by using such agencies is the increased risk of a c0ck up. I used to run a railway station and saw plenty of tickets, issued by those agencies, which weren't valid for travel on any train).
As Yorky Lass indicates, the train operators release their ticket allocations about 12 weeks in advance. (The regulator requires that they must make them available at least 12 weeks in advance. In practice, most operators wait until the last possible minute before doing so).
Yorky Lass has provided you with the only link that you need to check fares. If you can travel at off-peak times (such as lunchtime on a Tuesday) you'll probably find plenty of discounts available. However, it's almost certainly pointless looking for discount fares at peak travel times (e.g. on Friday evenings). In particular, many train operators don't offer any cheap fares between (roughly) December 20th and January 2nd.
Chris