I don't think「無念」(regret) and「無念無想」are related.
「無念無想」as already noted means free from worldly/worthless/distracting thoughts, and is Buddhist terminology. The Buddhist sense of「無念」is derived from this meaning.
大辞林
「むねん」
(1)〔仏〕種々の雑念を生む心を消滅させた状態。正念。
This is the only meaning common between Chinese and Japanese. The other meaning of「無念」in Chinese means to not forget, which is also counterintuitive at first sight, and Japanese regret certainly isn't derived from this meaning.
《詩・大雅・文王》:「王之藎臣, 無念爾祖;無念爾祖, 聿修厥德。」
「無念爾祖」here means don't forget (勿忘) your ancestors.
In fact, Japanese idioms containing「無念」meaning regret aren't found in Chinese at all, such as
so the meaning regret for「無念」is probably a Wasei-kango, unrelated to either the Old Chinese or the Buddhist appearances of「無念」.
Dai Kan-Wa Jiten explicitly defines「無念」as「残念」, and provides a Japanese Kanbun quote for this meaning:
{{ko:大漢和辭典}}
「無念」
(三)口惜しいこと。くやしいこと。殘念。〔東鑑、六〕文治二年四月八日、不見其藝{{ko:者}}、無念由。
「不見其藝者、無念由」> 其{そ}の藝{げい}を[不見者]{みずんば}、[無念]{むねん}の由{よし} (source)
Chinese doesn't (didn't before adoption from Japanese) have a word like「殘念」either, so it is also probably a Wasei-kango.
Since「無念」is a synonym of「残念」and are both probably Wasei-kango, then the logic of「無念」can be related to「残念」. Changing from「残」to「無」is analogous to changing from little remaining to none at all, and I suspect that this is what happened when the word「無念」was formed.