No. It might speed things up but if everything's in place it won't.
As for the market slowing down, it might be in places but in others it's certainly not slowing.
Having just gone through the sale and purchase process, I accepted a lower bid for my house because the prospective buyer wasn't selling a property. It shortened the chain and offered a bit more certainty.
Houses on the same estate have had sales fall through due to chains etc. Some on the market 5-6 months. Reasons vary, overpriced, structural or position.
The security of a cash buyer is not to be taken lightly. Needless to say best offer is still a goer so. Just observed so many sale agreeds put up and a few weeks later those stickers are peeled off...
That's it in a nutshell. Most sellers want to move on themselves. They'll happily sacrifice ~10k on a 750k property to get it all done and dusted within weeks rather than wait for months for a prospective buyer to sell their own place first.