Define one-parameter Lie transformation which takes values $\Psi(\varepsilon, x)$ to be a transformation which satisfies the following properties, where $\varepsilon \in I \subset \mathbb{R}$ and $x$ is in some suitable space to make it well-defined.
$$ \Psi(\varepsilon_1, \Psi(\varepsilon_2,x)) = \Psi(\varepsilon_1+\varepsilon_2, x) $$ $$ \Psi(0,x) = x $$
My question is whenever you have two such transformations, is it possible to create a two-parameter Lie transformation defined by the following properties, and so that two initial one-parameter Lie transformations are somehow included in the two-parameter transformation (for example, by taking $\Theta(\varepsilon,0,x)$ or $\Theta(0,\delta,x)$)?
$$ \Theta(\varepsilon_1, \delta_1, \Theta(\varepsilon_2, \delta_2, x)) = \Theta(\varepsilon_1+\varepsilon_2, \delta_1+\delta_2, x) $$ $$ \Theta(0,0,x) = x $$
Intuitively, I think it should be true as, for example, for groups, it is easy to define group on Cartesian product by $(g,h) \cdot_{G \times H} (g',h') = (g\cdot_{G}g', h\cdot_H h')$, but I do not see how this translates here. I also think that intuition for wanting this two-parameter group is to define transformation such that if we start from some $x$ then all $y$ that can be reached by finite applications of two one-parameter Lie transformations is the same set as all $y$ that can be reached by newly defined two-parameter transformation.
If possible, can you please provide example construction for $x = (x_1, x_2, x_3) \in \mathbb{R}^3$ where two one-parameter Lie groups are translation and rotation, as given next?
$$ \Psi_1(\varepsilon, x) = (x_1 + \varepsilon, x_2, x_3)$$ $$ \Psi_2(\varepsilon, x) = (x_1 \cos \varepsilon + x_2 \sin \varepsilon, -x_1 \sin \varepsilon + x_2 \cos \varepsilon, x_3) $$
EDIT: It seems that this example is not sufficiently interesting, as I believe it was worked out by user @Cosmas Zachos, and so I provide another one in hope to make it more interesting. These are just two rotations about different axes.
$$ \Psi_1(\varepsilon, x) = (x_1 \cos \varepsilon + x_3 \sin \varepsilon, x_2, -x_1 \sin \varepsilon + x_3 \cos \varepsilon)$$ $$ \Psi_2(\varepsilon, x) = (x_1 \cos \varepsilon + x_2 \sin \varepsilon, -x_1 \sin \varepsilon + x_2 \cos \varepsilon, x_3) $$
I really don't appreciate what you are asking. If you are asking for a simple representation or realization of your three maps, it is evident by inspection that $$ \Psi(\epsilon, x)= x+\epsilon, \\ \Theta (\epsilon, \delta, (x_1,x_2))= (x_1+\epsilon,x_2+\delta), $$ 1d and 2d translations; and, lastly, $$ \Psi_1(\epsilon,(x_1,x_2)) =(x_1+\epsilon, x_2) ,\\ \Psi_2( \epsilon, (x_1,x_2))= \begin{pmatrix} \cos \epsilon &\sin\epsilon \\ -\sin \epsilon &\cos\epsilon \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix} x_1 \cos \epsilon +x_2 \sin\epsilon \\ -x_1 \sin\epsilon +x_2 \cos \epsilon \end{pmatrix}, $$ 1d translation and 2d rotation respectively. I skipped $x_3$, as it is a superfluous inert parameter here, and need not be written down explicitly.
Response to EDIT in question. This it a composition of two rotations, a latitude and a longitude, $$ \Psi_1( \epsilon, (x_1,x_2,x_3))= \begin{pmatrix} \cos \epsilon &0&\sin\epsilon \\ 0&1&0 \\ -\sin \epsilon &0&\cos\epsilon \\ \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix} = \begin{pmatrix} x_1 \cos \epsilon +x_3 \sin\epsilon \\ x_2 \\ -x_1 \sin\epsilon +x_3 \cos \epsilon\\ \end{pmatrix}, \\ \Psi_2( \delta, (x_1,x_2,x_3))= \begin{pmatrix} \cos \delta &\sin \delta &0\\ -\sin \delta &\cos \delta &0\\ 0&0& 1 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix} = \begin{pmatrix} x_1 \cos \delta +x_2 \sin\delta \\ -x_1 \sin\delta +x_2 \cos\delta \\x_3 \end{pmatrix}, $$ non-commuting rotations about the y and z axes, respectively, so ε is an altitude angle an δ an azimuth.
You may always consider $\Psi_1(\epsilon, \Psi_2(\delta,\vec x ))\equiv \Theta(\epsilon,\delta, \vec x)$, via matrix multiplication of the above two matrices, to get a general rotation matrix, as you learned in college mechanics.