Jelly, 7 bytes

ṭ€Ðo⁾1*

A full program that accepts a string, as defined by the regex, and prints an executable expression.

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How?

Prepends each single character variable or single character constant with 1*.
Does not need to find these as they are guaranteed to be every other character.

ṭ€Ðo⁾1* - Main Link: list of characters (matching the regex in the question)
 €Ðo    - apply to each character at an odd index:
ṭ   ⁾1* -   tack to "1*"
        - implicit, smashing print

Also 7 bytes

⁾1*;ṛƭ)

Again a full program Try it online!.

Retina 0.8.2, 12 11 10 bytes

[^+]+
1*$0

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-1 porting l4m2 by accident while trying to think of something cooler

-1 thanks to Neil suggesting a version change

Fairly boring substitution, but probably not much that can outdo this here.

JavaScript (Node.js), 27 bytes

x=>x.replace(/\w+/g,'1*$&')

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