Is there mobile phone service while trekking in Nepal?

It depends on the route and the service provider generally NTC (Nepal Telecom) has better service in most remote areas, but sometimes the private provider Ncell is better such as in the Everest Region. There is a cell tower at Everest Base Camp. If you really want to do your best to maintain connectivity, you can pick up a SIM card for each network. SIM cards are cheap in Nepal and easy to get from any phone shop. You will need a copy of your passport and 2 passport photos in most cases.