Ah no no, wherever does this idea of “acid mushroom compost” come from!
Ph of compost is almost never an issue, no worries and I reckon yours is pH 6.8 to 8.5, depending whether they capped the mushroom beds with chalk.
I have used 8.5 compost on soil of pH 7,8, everything grew fine.
Yes your season is shorter, probably faster when it happens and your microbes.worms will get busy + be helped by the mulch.
Enjoy your winter, the ground is improving.