A second for Waddesdon although, like Neuschwanstein, it looks much older than it is. A very fine collection indeed.
Chartwell has the paintings (and studio) of Sir Winston Churchill, without whom many of us wouldn't be here.
Yes, the National Trust puts out an annual, colour, 432 pages in 2016, handbook telling about all of their properties. Much of the information is on line but combing through the handbook, arranged by area, can be much easier and very fulfilling. I'm sure that the Royal Oak Society in NYC would be able to source one for you.
Under the index of Collections subindexed as Paintings (other subindices are Armour, Carriages, Ceramics, Costume and Fashion, Dolls, Doll's-houses and Miniature Rooms, Furniture, Glass and Silverware, Musical Instruments, and Wall-hangings, Embroideries and Needleword) are listed 44 properties.
Particular favourites of mine that may fit your needs (and of course I haven't made a study, nor have I ever come anywhere near visiting all of the NT's properties despite being an active member (includes attending Annual General Meetings) for nearly 20 years are:=
Upton House and Gardens to show what an oil magnate could accumulate
Red House, Wightwick Manor and Gardens (and a whole stand of Redwoods), Nuffield Place, and Standen House and Garden - all of which are strongly associated with William Morris and the Arts and Crafts Movement
Petworth House for landscapes
Polesden Lacey
Ascott, just down the street from Waddesdon Manor
I guess that Fenton House, in Hampstead, North London, wouldn't meet your criteria for Manor House, which is a shame - it is one of my favourites, so easy to get to, such fabulous ancient (and playable) musical instruments, plenty of paintings, and other decorative arts
Ickworth House for its Rotunda and Gainsboroughs and Reynolds portraits
Knole
Then, not National Trust, but most definitely a Manor House - very nearly a palace - is Woburn House with its Canalettos.
For a start.....