Churchill had refused the Order of the Garter in 1945, quipping that he couldn't accept the garter from the king after the people gave him the boot. But in 1953 he relented and accepted the honour? Is it known what exactly made him change his mind?
The order of the garter was "Restored to gift of the Sovereign by Attlee in 1946". So maybe he wouldn't accept it from the Labour prime minister but would accept it from George VI.
I personally think sour grapes is the explanation, it fits with his personality, which seems a bit tempestuous. And he did change party (or "Cross the Floor") twice, so perhaps he was open to changing his mind from time to time.
What makes the most sense to me was that in 1945 he (actually his party) had just been voted out of office. At this point he still had hopes of getting back the majority (and perhaps the PM office), and in fact he did regain it in 1951.
So I think in 1945 he was mostly telling you what his problem was. He was still an active politician, and as such it would be next to impossible to accept such an honor without it being tainted by Great Britian's extant political climate. People could not help but speculate that there were crass political dimensions to the "honor".
In 1953 he was PM again. However, he was also 78, and suffered a fairly bad stroke, after which he never walked or spoke quite right again. He was clearly in his declining years, and in fact retired the PM position 2 years later. So at this point the logic that compelled him to decline the honor a decade prior no longer applied. Its also quite possible that, given his failing health, the soverign was a bit more insistent on giving it to him this time, for fear of any further wait causing it to have to be bestowed posthumously.