Charcoal, 34 bytes
⊞υ⮌ＳＦISBH42≔ΣＥυ…⊞ＯＥ⪪κι∨μＩν0²υＩ⍘⮌υ² Attempt This Online!Attempt This Online! Link is to verbose version of code. Will produce values up to 18446744073709551615. Explanation:
⊞υ⮌Ｓ Start with the reverse of the input in a single piece.
ＦISBH42 Work down from the highest double power of two to the lowest.
≔ΣＥυ…⊞ＯＥ⪪κι∨μＩν0²υ Split each piece into two, defaulting the first piece to 0 and the second piece to 1, but if the split only resulted in one piece then pad it with a 0 piece. Examples: 2 splits on 2 to give two empty strings, so the first becomes 0 and the second 1, representing 10 in binary; 21 splits on 2 to give 1 and an empty string, the latter becoming 1, representing 11 in binary; 1 splits on 2 to give 1, which is padded with a 0, representing 01 in binary.
Ｉ⍘⮌υ² The reversed list is now a list of binary digits so convert it to decimal.
A port of @Pufe's Ruby answer is also 34 bytes:
≔⁰θＦＥＳＸ²÷Ｘ²⌕124HBSIι²≧⁺∨×﹪θι⊖ιιθＩθ Try it online! Link is to verbose version of code. Explanation:
≔⁰θ Start with zero.
ＦＥＳＸ²÷Ｘ²⌕124HBSIι² Map the characters of the input string to the double powers of 2 and loop over the results.
≧⁺∨×﹪θι⊖ιιθ Add on the current value modulo the current power multiplied by one less than the current power, unless that is 0, in which case just add on the current value.
Ｉθ Output the final result.