While many of my answers in the 'Travel' section owe more to to googling than to personal knowledge, this one definitely falls within my area of personal knowledge. (i used to run a railway station).
My first bit of advice is to avoid all of the so-called discount websites. All that they do is offer you exactly the same fares that you can get through the National Rail website but then they often add a 'booking fee' on top. This is the only link which you should be using:
http://www.nationalrail.co.uk/
My next point is that it might be too late to get any discounted 'Advance' fares for the Bank Holiday. (For the best fares you need to book around 6 weeks in advance. Some rail operators withdraw all discounted tickets on Bank Holidays anyway).
If the tickets you're offered on the National Rail website are described as 'Advance' then you MUST travel on the train service specified on the ticket and, if you're given reserved seats, you MUST occupy those seats only. Otherwise you'll find that ticket inspectors will treat the situation as if you'd boarded the train without any ticket at all. That means that you'd have to pay the maximum standard single fare, possibly with a penalty fare as well.
However if the tickets offered on the National Rail website are 'Off Peak' or 'Standard' fares, they're exactly the same as the the 'walk up' tickets you could have purchased at the station, with the same validity on trains.
Chris