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[OED:] Etymology: For forth with (preposition), = earlier forth mid, along with, see forth adv. 2c. The adverb forthwith originates from this phrase, the preposition being used absol. or with ellipsis of its regimen.

[forth adv. 2c] †c. In early Middle English forth mid, later forth with = ‘along with’. Also absol., along with him, them, etc.: at the same time with something else.

at the same time ... does not mean 'immediately'. Per this answer, with can mean 'with' or 'against'; so I don't know how to disambiguate it.
Here's my guess at the etymology: Suppose you want to proceed forth with X. Then X may need be completed immediately. Please allow me to verify, to avoid deluding myself with false etymologies.

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  • i think it's simply leaving off the final of the three words. so it was originally someting like "forth with urgency" or "forth with vigour". As it says, it is short for "forth with (preposition)". Also, you're familiar with the form "go forth with pride.." "go forth with a smile"
    – Fattie
    Commented Jun 5, 2015 at 3:15

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This very query has just been answered by World Wide Words. Here's an extract:

Round the middle of the twelfth century, the phrase forth mid appeared (mid being essentially the same as the modern German word mit, with), later forth with, to go somewhere in the company of other people. Necessarily, if you go forth with others, you go at the same time as they do. It seems this sense of time eventually took over, though the process of transition isn’t very clear, and it’s mixed up with other phrases that also referred to time. Certainly, by about 1450 the phrase had condensed to a single adverb with the modern meaning of immediately, without delay.

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  • Thanks. But is there more info or some way of explaining the following from your answer? It seems this sense of time eventually took over, though the process of transition isn’t very clear, and it’s mixed up with other phrases that also referred to time.
    – user50720
    Commented Jun 5, 2015 at 14:51
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    @LawArea51Proposal-Commit Transformations like that just happen naturally because people conflate related concepts. How can you ask so many questions about etymology and not realize this yet?
    – Barmar
    Commented Jun 5, 2015 at 19:55