If you have already decided to purchase a home and you don't have the luxury of waiting for a market change, your lending rate will in large part depend on the market. If you have time to decide when to jump into the mortgage market, then research mortgage rate forecasts for the next year. If rates are expected to increase you may want to submit an application quickly, if rates are moving down you can wait.
Of course your credit rating will be another relatively fixed variable in your quest to get the best rate, if it's relatively low you will pay a higher rate. So if you have time before you need to borrow attempt some credit repair through one of the better known repair firms. It usually takes a few months for them to make a difference but the points they gain could save you significantly over the term of your loan.
It's common sense to shop around for the best mortgage rate and terms and negotiate with mortgage lenders until you are satisfied that you have been quoted the best available rate. So, you can go through whatever expedited process you want, you need to establish to best rate and terms before you zoom through to closing.
Mortgage companies may lock in your interest rate once you apply and are approved but if rates are too often changing they may not offer you an opportunity to lock the rate.
As an appraiser and I understand how important your valuation is. If you have problems with the appraisal the rest of the transaction can just fall apart. Address valuation issues before you bother with loan commitments, if the property you want to finance won't appraise as high as expected you may be wasting your time.
If you do get a commitment of any kind from a mortgage company make sure you get a copy of it, some mortgage companies have gone to on-screen signature documents that disappear once you have electronically signed them. The document you need may disappear when the mortgage lender no longer want to honor them at closing, it's happened to me, so you better have a "screen print" copy before your disappears into the ozone.
Historically mortgage rate have varied between 3.0% and 18.0% and at the moment, in early 2018, the 30-year fixed mortgage rate is near 4%. So we have current mortgage interest rates that are near the low-end of the mortgage interest rate range and if you wait no guarantee exists that they will remain low indefinitely.
If you take the time to make sure the home you want to borrow on will appraise near your expectations, that you are in the best credit position to borrow, you have shopped for the best available rate / terms and considered timing you will get a lot more for your money.