I can't think of a single year in recent history where Heathrow Express hasn't operated an inter-terminal shuttle over Christmas when the rest of the railway is not operating (Christmas Day/Boxing Day) or engineering works. The last occasion there was no train service in the tunnels due to engineering work they operated a replacement bus shuttle even on Christmas Day.

(In the interests of disclosure I probably should point out that I am the proprietor of the railway website mentioned upthread.)