Disintegration theorem, a reference needed

The most comprehensive account of disintegration can be found in David Fremlin's magnum opus Measure Theory in Chapter 45 in Volume 4.1. But this is certainly overkill.

The state of the art paper is Disintegration and Compact Measures by Jan Pachl, he gives a characterization result and nothing more general is possible. His approach is based on the von Neumann-Maharam lifting theorem and this approach to disintegrations was pioneered by Hoffmann-Jorgensen in Existence of Conditional Probabilities. The approach based on liftings has the advantage that it needs no separability conditions, the cost is that the disintegrations are only measurable with respect to a completion.

Under separability assumptions, there are necessary and sufficient conditions known for certain kinds of disintegrations. You can find very useful results to this effect in a beautiful paper by Arnold Faden: The Existence of Regular Conditional Probabilities: Necessary and Sufficient Conditions.

Now these paper represent the high end of mathematical probability theory. For most applications, one can use much more elementary methods. For conditioning on $\mathbb{R}$, the book by Lehmann and Romano already mentioned gives a very readable proof under Theorem 2.5.1 (2005 ed.). In Billingsley's, Probability and Measure, 3rd ed, you can find the same result in section 33 as Theorem 33.3. This results are more powerful than it may seem at first. By a famous isomorphism theorem, every uncountable, separable and complete metric space endowed with the Borel $\sigma$-algebra is isomorphic as a measurable space to $\mathbb{R}$ with the Borel $\sigma$-algebra. If you want to see a proof of the strong version of the result without using this isomorphism theorem, you can check Theorem 10.2.2 in Dudley's Real Analysis and Probability (2002/2004 version). The proof in Dudley is considerably harder.

Chapter 6 of Foundations of Modern Probability (Second edition) by Olav Kallenberg starts with the sentence

"Modern probability theory can be said to begin with the notions of conditioning and disintegration."

I think you will find the whole chapter a useful reference.

A nice book on Measure Theory by Bogachev deals with disintegration in Chapter 10 (Volume 2).