National Rail said travellers should plan for some services to be cancelled at short notice.
It said the issue appears to be a fault with the onboard GSMR radio system, which is used to communicate in case of emergencies.
According to the Rail Safety and Standards Board Rule Book, in the case of a radio system failure, trains cannot exceed 100mph (160km/h), or 60mph (100km/h) through affected areas.
National Rail said it is investigating the cause of the fault. It advises that passengers may be entitled to compensation if their journey is delayed.
Across the network, most delays appear to be no longer than 15 minutes, but some services have been delayed be up to an hour.
National Rail warns that there are delays of around 10-15 minutes are expected on Southern, Thameslink and Great Northern services, with some short notice cancellations possible.
Most passengers on these lines will be able to travel on their normal route, National Rail has said. Disruption is currently expected to continue until 12:00 GMT.
The Gatwick Express is only running between London Victoria and Gatwick Airport, not serving Brighton, while some Heathrow Express trains between Reading and Heathrow Airport are also delayed.
Elizabeth line services on this route are also affected.
Trains from Brighton to Southampton Central will now only be running to Bognor Regis or Chichester. And services between Kings Lynn and London Kings Cross will only run between Kings Lynn and Cambridge.
ScotRail says its services were impacted earlier this morning but are now operating normally.
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