Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?

I want to know whether there is some general assumpitons we can make on two measurable spaces $$E$$ and $$F$$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" holds:

Theorem: Suppose that $$X: \Omega \rightarrow E$$ and $$Y: \Omega \rightarrow F$$ are two random variables on a probability space $$(\Omega, \mathcal{F},P)$$. Then there exists a probability space $$(\hat{\Omega}, \hat{\mathcal{F}}, \hat{P})$$, random variables $$\hat{X}:\hat{\Omega} \rightarrow E$$ and $$\hat{U}: \hat{\Omega} \rightarrow [0,,1]$$ with $$\hat{P} \circ X^{-1} = P \circ X^{-1}$$ and $$\hat{U}$$ uniformly distributed in $$[0,1]$$, and a product-measurable mapping $$\phi : E \times [0,1] \rightarrow F$$ such that $$P \circ (X,Y)^{-1} = \hat{P} \circ (\hat{X}, \phi(\hat{X}, \hat{U}))^{-1}.$$

I saw something similar when proving existence of discrete time markov chains, where $$E$$ was assumed to be at most countable and endowed with the discrete $$\sigma$$-algebra. And I also came across this paper by Skorohod which assumes (in the above setting) that $$E=F$$ is a Polish space and proves an even stronger result then the above. But what general assumptions can we make about $$E$$ and $$F$$ such that the above can hold?

Thanks a lot in advance!

Lemma. Let $$(\Omega,\mathcal{A},\mathbb{P})$$ be a probability space, $$X$$ a real-valued random variable on this space, and $$\mathcal{C} \subset \mathcal{A}$$ a $$\sigma$$-field. Let $$F$$ be the cumulative distribution function of the conditional law $$\mathcal{L}(X \mid \mathcal{C})$$. Let $$\xi$$ be a random variable with uniform law on $$[0,1]$$ and independent of $$\mathcal{C}\vee\sigma(X)$$. Define $$U = F^-(X) + \xi\bigl(F(X) - F^-(X)\bigr).$$ Then $$U$$ is a random variable independent of $$\mathcal{C}$$, uniformly distributed on $$[0, 1]$$, and one has $$X = G(U)$$ where $$G$$ is the right-continuous inverse function of $$F$$.
Proposition. Let $$(\Omega,\mathcal{A},\mathbb{P})$$ be a probability space. Let $$\mathcal{C}\subset\mathcal{A}$$ be a $$\sigma$$-field, $$V$$ be a real-valued random variable, and define $$\mathcal{B} = \mathcal{C}\vee\sigma(V)$$. Then there exist an embedding $$\iota$$ from $$(\Omega,\mathcal{A},\mathbb{P})$$ to a probability space $$(\hat{\Omega},\hat{\mathcal{A}},\hat{\mathbb{P}})$$ and a uniform random variable $$\hat U$$ independent of $$\hat{\mathcal{C}}$$ such that $$\hat{\mathcal{B}} \subset \hat{\mathcal{C}} \vee \sigma(\hat U)$$.
Apply this proposition to $$\mathcal{C} = \sigma(X)$$ and $$V$$ real-valued such that $$\sigma(V) = \sigma(Y)$$, which exists assuming $$E_2$$ is a standard Borel space.