Great Northern and Anglia are both a doddle with a bike, as long as you are aware of which services have restrictions - no pre-booking needed. There aren't any dedicated bike spaces on the trains, but still, works well enough. Generally, as my bike-on-a-train journeys only tend to go as far as Cambridge, so the restrictions usually don't apply.
My one experience of SWR is also pretty positive. Dedicated bike spaces on the train, again, no pre-booking needed, and bike spaces are in the quiet carriage, which is an added bonus.