"Does non-financing contingency expedite the process? "

Not enough to matter.

The big delay will be getting the note holder to sign off.

Banks and lenders do NOT like to lose money.

Make the offer and sit back and wait to hear from them. Cash will not speed up the process, but it is more appealing than an offer with financing contingencies.

Best of luck - I think you are going to need it. The chances of getting a short sale through with 2 lenders is pretty small.

Just by the numbers, if a house is worth $450k, why would the first mortgage holder take less than $450k? You might have a little wiggle room for the convenience of getting it off their books, but not much. Your $400k offer is going to net about $375k, so they will be losing $75k. That isn't even counting the second mortgage. Obviously, the second lender isn't going to approve unless they get something, so that loss to the first actually has to be larger.

It is possible that they will accept your offer, but I don't think it is likely. The first lender is better off just foreclosing and trying to sell at market value.

"1 loan 450K, the other 50K."

The second mortgage holder is likely to foreclose since they will be wiped out if the first forecloses before they can.

Unless they want to lose the entire second they may step up and take the house.

If it is worth less than the first they may view the loss as better than taking on the first mortgage liability.

This is why seconds get a higher rate than firsts.

You are looking at a long path with two lenders, either of whom can kill the deal.