You received an entry clearance with a start date of 7 Aug, but a change in plans means you will arrive in the UK on 31 July. You want to know if this is OK?
No. It's decidedly not OK. They do not like it when this happens and so they have a rule in place to cover it. It's Paragraph 30C, which says...
An Immigration Officer may cancel an entry clearance which is capable
of having effect as leave to enter if the holder arrives in the United
Kingdom before the day on which the entry clearance becomes effective
or if the holder seeks to enter the United Kingdom for a purpose other
than the purpose specified in the entry clearance.
They have the authority to vary the dates on the spot, but this requires an awesome level of personal impact and articulation skills, we're talking personal impact like Kim Kardashian here. So for practical purposes you can expect to be in trouble (in the very strange event that the carrier will allow you to board in the first instance).
What can I do right now?
You need to go through a VFS even though you did not go through one originally; they are the only way a person can access priority services. Contact whichever VFS can act as your liaison in this matter and sign up for one of their super priority services. Since you are US based there's a wealth of them to select from, you can even customise one if you have the money. There is still enough time between now and your flight to get a variation on your visa's start date.