In ZFC, is there an uncountable atomless Boolean algebra $B$ such that for all countable subalgebras $A_1,A_2\subset B,$ every isomorphism $f:A_1\to A_2$ extends to an automorphism of $B$? $A_1$ and $A_2$ can be finite.
Here is some background. Let's say a first-order structure $\mathfrak B$ with domain $B$ is $\kappa$-ultrahomogeneous if for all $\lambda<\kappa$ and all tuples $a,b\in B^\lambda$ with the same type over $\mathfrak B,$ there is an automorphism $f$ of $\mathfrak B$ with $f(a_i)=b_i$ for all $i.$ So I'm interested in $\aleph_1$-ultrahomogeneity. But the only method to construct even an $\aleph_0$-ultrahomogeneous structure that I know of is to take a saturated model. Let $T$ be the first-order theory of infinite atomless Boolean algebras. For infinite $\kappa,$ there is a saturated model of $T$ of cardinality $\kappa$ if and only if $\kappa^{<\kappa}=\kappa.$ Under certain assumptions it is consistent that this does not occur.
(A Boolean algebra $B$ is usually called "homogeneous" if for every $a\in B\setminus\{0\}$ the algebra $B/(1-a)$ is isomorphic to $B.$ This is different again from the model theoretic homogeneity as defined in Chang & Keisler's book.)
There are some other standard examples of complete $\omega$-categorical unstable theories, but I don't know if they have $\aleph_1$-ultrahomogeneous models either. Infinite dense linear orders without endpoints, random graph/digraph/hypergraph, vector spaces over a fixed finite field equipped with a symplectic/orthogonal/hermitian form. There is at least an uncountable $\aleph_0$-ultrahomogeneous dense linear order: the order type of $\mathbb R,$ using piecewise linear maps for $f.$
GCH is an acceptable assumption for proving arithmetic statements. But model theorists do sometimes care about removing the GCH assumption. See for example Chapter 5 "Saturated and Special models" of Chang & Keisler, Model Theory, 3rd Ed. I didn't find any results for this $\kappa$-ultrahomogeneity property.
Best Answer
The property you call "$\kappa$-ultrahomogeneous" is usually called "strongly $\kappa$-homogeneous". It's a theorem of ZFC that for any structure $M$ and any infinite cardinal $\kappa$, $M$ has an elementary extension that is strongly $\kappa$-homogeneous. In particular, your question has a positive answer by taking a strongly $\aleph_1$-homogeneous elementary extension of an uncountable atomless Boolean algebra. See this question and my answer for references to the proof.