Judging whether prices have risen or fallen under privatisation is tricky as the pricing structure has become much more complex. There are a rash of new ticket types and lot of new discounted fares available. The best discounts tend to be available from booking in advance, often by a week or more.
Pricing depends to a great extent on the elasticity of demand for a product.
In the case of train travel there are a wide variety of users with different elasticities. People who travel at peak times for commuting or business travel will tend to have much more inelastic demand curves for train travel as they need to get to their destination for work or other reasons. An inelastic demand curve is shown below. If demand is inelastic then users can be charged a higher price without there being a significant fall in the level of demand. Hence higher prices for morning and afternoon rush hours and those who just turn up for a train.