I think it's likely the Elizabeth Line branch is suspended because the stations are closed, and the stations are closed because there's no power to the buildings.
Other trains are running on the main line through the area. It's 25kV electrification, there's plenty of distance between feeder substations.
National Rail say [1] "An unadvertised shuttle service is running from Heathrow Terminals to London Paddington to ensure customers and colleagues can leave the airport" so trains can and are running, they just don't want flight passengers to go to the airport.