Origin of British term "to bits"
Solution 1:
A Dictionary of Slang and Unconventional English gives thrilled to bits "since late 1940s, perhaps ob[solete] by 1970", though a search of Google Books gives references in the 1930s.
"Chuffed to bits" also depends on the prevalence of "chuffed", of course; I was able to find one example as early as 1979.
Nowhere have I found any authoritative reference to the source of the to bits phrasing, though John Lawler's suggestion sounds reasonable.
Solution 2:
It may come from the older expression "(all) to pieces: to a great degree, completely, through and through. Now colloq." (OED)
Doesn't help directly with the etymology, but the first citation, from 1788, is "It beats Pinetti's conjuring all to pieces", suggesting an extension from literally beating into pieces, to "he beat me to pieces at chess", to "I love you to pieces (Snookums)".
Solution 3:
Ngram shows that the expression love to bits is quite recent:
Love to bits:
love somebody/something to bits (informal) to like or to love someone or something a lot.
- Clive's the nicest person I know. I love him to bits. 'Do you like your new bike, then?' 'Oh, I love it to bits!'
Source: Cambridge Idioms Dictionary.
but the expression 'to pieces' ( also love to pieces) can be found in much older citations.
here are the Oxford English Dictionary's relevant citations (in the sense "to a great degree" and not necessarily literally "to ruin" etc.):
1788 ‘P. PINDAR’ Prooemium in Wks. 366 'Tis most extraordinary then, all this is—It beats Pinetti's conjuring all to pieces. //
1840 C. F. HOFFMAN Greyslaer I. I. x. 114, I know the ground here all to pieces. //
1892 W. G. LYTTLE Ballycuddy 79 She wud a pleesed ye a' tae pieces, an' wud a been charmed tae a haen a minister fur a son-in-law. // 1925 Dial. Notes 5 325, I knows un all to pieces. //
1958 L. URIS Exodus III. iii. 351 Ari! I love you to pieces! // 1989 B. A. MASON Love Life 123, I just love him to pieces.
Source: http://en.m.wikipedia.org