For the benefit of the OP and anyone else who might be interested, do check out TfL's timetable planner: Tube, Overground, TfL Rail, DLR & Tram status updates - Transport for London

it'll provide info on when the Piccadilly line to LHR (and all other lines for that matter) aren't operating on weekends. For instance, on 12 and 13/2 there'll be reduced service on TfL Rail (Crossrail) between LHR and Paddington, then again on the 19th and 20th and on the 26th. Don't know if it'll affect HEX too.