We just went from London to Swindon and it cost us 44 euro each one way. If we had known to buy those tickets the night before, the cost would have been about one third.
Jan is right, always by tickets day before. Also,
check before hand if ticket is for Sunday because
that is day they do routine rail maintenance.
Since you mentioned it, Arn, is it really a big hassle to travel via train in England on Sunday? I'm arriving at Heathrow around 1pm on a sunday in late July and need to get to stratford-upon-avon... I hope it's not TOO big a hassle, eek.
Even better- use a BritRail Pass. They come in several combinations (3 out of 8 days, 8 out of 15 or 30, etc....). We love the flexibility.