Any thoughts from locals on how hard it will be to travel through London on Coronation Day?
Coming in on Eurostar and traveling on to Liverpool, same day.
I was wondering if I should try to travel a day early.

You shouldn't have any problem (assuming you'll have your tix early) because the Eurostar arrives at St Pancras and the Liverpool train leaves from just half a mile down the road at Euston Station. Easily walkable. You won't be anywhere near the Westminster / central areas mostly impacted by the coronation festivities.

Agree. Both stations will be packed so try to arrive as early as possible.

^^^ - ?? The OP is arriving on the Eurostar into St Pancras so won't have choice when to arrive at the station -- s/he will get there when the train does.

The OP could take an earlier Eurostar.