All of the English lyrics are modifications of the German, there is no 'correct' translation.The arrangers have attempted to fit English to the melody, and the German words written by Brahms, and have had to depart from any straight literal translation.
Most people have no idea of the problems faced by translators of poems or songs- word for word translation is impossible. See Andreas W above.   

The bedacht to bedight transfer appears in the English lyrics of 1873 (and 1868), by Auber Forestier, linked by Masato above; most singers have followed. I can't think of a better solution that also fits the notes of the melody.
The words following 'bedacht' are- "mit Näglein besteckt schlupf' unter die Deck'"- this does not translate as "Creep into thy bed, There pillow thy head." Closely translated, it is- garnished (clad) in pinks (nightdress), sleeping under the covers. The English words get the idea of sleeping across although the covers and the baby's night dress get left out and a bed is introduced.
Note that I am translating 'näglein' as pinks, an accepted meaning in one of my German-English dictionaries, and guessing that it would have been an idiom for a baby's night clothes when the song was written. I also assume that näglein is an alternate spelling. These assumptions may be in error, but without an idiomatic German dictionary for the 1860s, I can't be sure.

"Wiegenlied" is one of five solo songs in Brahms' Opus 49. He wrote many songs; 125 were published in groups of four to nine.