The union said members at both companies were solidly supporting the industrial action.
The SWR strike is expected to impact services in and out of London’s Waterloo station, which is one of the capital’s busiest hubs and is used by many of the thousands of spectators who attend the fireworks display at the nearby London Eye.
Tens of thousands of people are expected to attend Edinburgh's Hogmanay celebrations tonight, with organisers saying the event - one of the world's biggest street parties - will be unaffected by the arrival of Storm Dylan earlier today.
North Korea’s Pyongyang display outshone that of Japan half an hour earlier.
But both were dwarfed by a 10-minute extravaganza on Hong Kong’s Victoria Harbour.
London Ambulance Service (LAS) said it took 7,000 emergency calls on 31 December last year, 2,000 more than a typical day.
Extra control room staff, ambulance crews, and support workers have been drafted in to cope with the increase in demand tonight.
Police in London will be employing vehicle barriers at tonight's New Year's Eve celebrations.