If you mixed in an appropriate source of Nitrogen those leaves would be digested by bacteria and converted into compost. Mushroom compost would have some N but probably not enough to finish composting those leaves by spring.
Perhaps the link below might be of some help.

Here is a link that might be useful: Compost Tutorial

Or take advantage of all of the free UCG's from the many coffee shops in Washington. That might be a good source of N for all of those leaves.

Mixing compost with a ingredient does not a whole lot of sense. Use the shredded leaves to make a compost pile. Both of my large piles consisted partly of maple and sweet gum leaves, which almost self-compost. The more oak leaves I use, the more greens needed for a hot pile.

Either wait until grass is available in the spring, or find another 'green' material such as used coffee grounds or fruit and veggie waste. It is easier to find grounds than a grocery that will provide fruit and veggie discards.

As to digging leaves into the ground, how finely shredded the leaves start out as is one factor on how quickly they break down. Maple and sweet gum break down quicker than oak, but oak is a staple of mine.

May be a bit like composting a 150 pound deer. Takes a year properly composted in a pile on top of the ground. Takes a number of years when buried. Not that I expect your in-ground composting to take that long.

What would happen over winter if I mixed them with an equal amount of mushroom compost. Thank you in advance
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1. IMHO, if you turn the pile every week or two, it will break down.

2. Use the leaves as mulch over winter. Come spring time, just sorta pull back the leaves in the spots you plan to grow.

3. Bag em up until you get some grass clippings next spring.

I added 12 pickup truck loads of leaves to my 2000sq foot garden last december. BY Febuary I was putting potatoes in the ground. I planted all the other vegetables in April. By June the leaves were over 90% broke down. The secret is choping the leaves up with a lawn mower and tilling them in. The bacteria in the groung should do its job as long as there is plenty of moister. Be sure to have soil test done regularly because leaves can change the ph drastically. Any leaves you have left over either pill them up for next year or I recommend chopping them up fr mulch around your plants. If you live in a high termite area only use cypress mulch around your house, or ceder!!!