I've used GWR Bristol to Didcot, just turned up, no issue.
Southend, Brighton to London same story even though as it was an unusual pleasant day in September and the number of bikes on the train far exceeded the stated capacity.
Chiltern rail - always found the bike space available.
Only time I have ever heard of an issue is when my colleague wanted to go from Brighton to lindon in morning rush hour. He did get on, but fellow passengers were not impressed as the train filled up.