As Dzug has stated, times vary with National Rail services. For trains terminating in London, the rule is usually that off-peak fares apply to all trains with an arrival time after 1000. This means that the cut-off for off-peak fares tends to be earlier the further away you are from London.
Off-peak fares for most local services in East Anglia start at 0845. Elsewhere in the country they tend to start either at 0900 or 0930.
Remember, if you're travelling to or from London, that (unlike the London Underground) peak time fares also apply to the 'evening peak'. On some lines this period can start as early as 1500. On most lines, however, the eveing peak starts at either 1600 or 1630. The evening peak ends, on almost all lines, at 1900.
Chris