Welcome to the forum simonp.
The 'Network Area' easement doesn't apply to your return from Yatton to Cambridge at the time you are travelling. It would apply if you were returning from Yatton in the afternoon and hit the evening peak out of Kings Cross or Liverpool Street to Cambridge. You'd be allowed to travel on an evening peak service to Cambridge.
As for the earliest return train you can catch from Yatton, looking at
restriction YC and the list of stations and the times you can travel from them, it would appear there is an error as Yatton is missing from the list. Worle is there (valid after 0750) as is Nailsea & Backwell (valid after 0800).
Because of that omission you'd have to follow the 'rule':
Travel from stations not listed above: Off-Peak tickets from stations not listed are valid for connections into trains departing as shown above.
Which would mean you can take any service from Yatton that connects, most logically, at Bristol Temple Meads for a departure from there after 0815 as per the restriction. That would be the 0830 to Paddington. As there are no rules on maximum connection times I'd suggest you can take any train from Yatton to connect into a service at another station listed under restriction YC, as long as it's on a permitted route of course.
As per my reading of the restriction you would be permitted to travel on the 0637 from Yatton, but only to then connect onto a valid service further up the line, which means a long wait at Bristol Temple Meads, Bristol Parkway, Bath Spa, Chippenham or Swindon!
BUT, if National Rail Enquiries is giving you an itinerary from Yatton at 0637 through to Cambridge on the return portion of your Off Peak Return, then your ticket is valid.
NRE‡ is the definitive public source for fares and validity information. Also, if a ticket retailer is offering travel on the 0637 from Yatton on the Off Peak Return then that is also valid. Advice in these circumstances is to print your itinerary as proof that NRE and the retailer have given you those trains to travel on. As you've done in the past, go for seat reservations as well.
The error in the National Rail journey planner is probably as a consequence of the omission of Yatton from restriction YC.