I would say that the result is implicit in Serre's Lie Algebras and Lie Groups. In LG IV and V he proves the categorical equivalence between formal groups and Lie algebras over any field of characteristic 0. He also remarks (end of Section V.4) that when the Lie algebra is nilpotent, the formal group is just a polynomial, so there are no convergence issues. [Edit: somehow I didn't notice until after I wrote this that this is almost exactly what you said above. If what you wrote was the sketch of the proof that you wanted to see in print, then I think I have answered your question. If however this was an example of how various texts include all the essential ingredients but never seem to tie it all together: sorry, I've just underscored your point.]He (almost) surely doesn't state the result explicitly since he doesn't talk about algebraic groups per se, but I think that stringing together the big theorem and the comment is an acceptable, albeit not ideal, reference.
By coincidence, I have Hochschild's book checked out of the library, so I tried to look up the result in it. Not much luck -- indeed it is technical (even?) compared to most other books on linear algebraic groups, not so well indexed, and uses some nonstandard notation. (I have little doubt though that if I could read it from cover to cover my understanding of the subject would be greatly enriched.)