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Consider some non-Borel set $Y \subset [0,1]$ (e.g. Vitali set).

Enumerate $Y$ using ordinals as $Y=\{x_\alpha\}_{\alpha < \beta}$. Let $m$ denote the smallest ordinal such that $X=\{x_\alpha\}_{\alpha < m}$ is non-Borel. Note that $m\ge \omega_1$, since otherwise $X$ would be at most countable.

Then the family $\mathfrak A = \{A_\gamma\}_{\gamma<m}$, where $A_\gamma = \{x_\alpha\}_{\alpha<\gamma}$, has the properties desired in the OP, but $\bigcup_{A \in \mathfrak A} A$ is non-Borel.

For more details about ordinals see e.g. Set theory by T. Jech (2006).

Construction

Consider some non-Borel set $Y \subset [0,1]$ (e.g. Vitali set).

Enumerate $Y$ using ordinals as $Y=\{x_\alpha\}_{\alpha < \beta}$. Let $m$ denote the smallest ordinal such that $X=\{x_\alpha\}_{\alpha < m}$ is non-Borel. Note that $m\ge \omega_1$, since otherwise $X$ would be at most countable.

Then the family $\mathfrak A = \{A_\gamma\}_{\gamma<m}$, where $A_\gamma = \{x_\alpha\}_{\alpha<\gamma}$, has the properties desired in the OP, but $\bigcup_{A \in \mathfrak A} A$ is non-Borel.

For more details about ordinals see e.g. Set theory by T. Jech (2006).

Discussion of cardinality

Let us prove that in fact $|X| = \aleph_1$.

Indeed, by construction $\aleph_1 \le |X| \le 2^{\aleph_0}$. Any uncountable Borel set has cardinality $2^{\aleph_0}$ (e.g. by Theorem 13.6 in Classical Descriptive Set Theory by A.S. Kechris or Corollary 2C.3 in Descriptive Set Theory by Y.M. Moschovakis.). Therefore, if CH fails then $|X|\le \aleph_1$, since otherwise there would exist a Borel subset with cardinality $\aleph_1$ (by minimality of $m$). On the other hand if CH holds then immediately $|X| = \aleph_1$.

Most of this answer comes from the very useful comments in this thread.

One can expand the comments by François G. Dorais and James Hanson using ordinals.

First, assuming weak CH, note that there exists a non-Borel set with cardinality $\aleph_1$. Indeed, take some set $X \subset [0,1]$ such that $|X| = \aleph_1$. The cardinality of the power set $\mathscr P(X)$ is $2^{\aleph_1}$, while the cardinality of the Borel sigma-algebra $\mathfrak B(\mathbb R)$ is $2^{\mathfrak \aleph_0}$, see e.g. here. Hence there exists $A \in \mathscr P(X) \setminus \mathfrak B(\mathbb R)$.

Next one can enumerate $A$ using ordinals as $A = \{x_\alpha\}_{\alpha \in \omega_1}$. For any $\alpha \in \omega_1$ let $A_\alpha := \{x_\beta : \beta \in \alpha\}$. Then $A_\alpha$ is countable and hence Borel. Moreover, for all $\alpha,\beta \in \omega_1$ either $A_\alpha \subset A_\beta$ or $A_\beta \subset A_\alpha$.

For more details about ordinals see e.g. Set theory by T. Jech (2006).

Consider some non-Borel set $Y \subset [0,1]$ (e.g. Vitali set).

Enumerate $Y$ using ordinals as $Y=\{x_\alpha\}_{\alpha < \beta}$. Let $m$ denote the smallest ordinal such that $X=\{x_\alpha\}_{\alpha < m}$ is non-Borel. Note that $m\ge \omega_1$, since otherwise $X$ would be at most countable.

Then the family $\mathfrak A = \{A_\gamma\}_{\gamma<m}$, where $A_\gamma = \{x_\alpha\}_{\alpha<\gamma}$, has the properties desired in the OP, but $\bigcup_{A \in \mathfrak A} A$ is non-Borel.

For more details about ordinals see e.g. Set theory by T. Jech (2006).

One can expand the comments by François G. Dorais and James Hanson using ordinals.

First note that there exists a non-Borel set with cardinality $\aleph_1$. Indeed, take some set $X \subset [0,1]$ such that $|X| = \aleph_1$. The cardinality of the power set $\mathscr P(X)$ is $2^{\aleph_1}$, while the cardinality of the Borel sigma-algebra $\mathfrak B(\mathbb R)$ is $2^{\mathfrak \aleph_0}$, see e.g. here. Hence there exists $A \in \mathscr P(X) \setminus \mathfrak B(\mathbb R)$.

Next one can enumerate $A$ using ordinals as $A = \{x_\alpha\}_{\alpha \in \omega_1}$. For any $\alpha \in \omega_1$ let $A_\alpha := \{x_\beta : \beta \in \alpha\}$. Then $A_\alpha$ is countable and hence Borel. Moreover, for all $\alpha,\beta \in \omega_1$ either $A_\alpha \subset A_\beta$ or $A_\beta \subset A_\alpha$.

For more details about ordinals see e.g. Set theory by T. Jech (2006).

One can expand the comments by François G. Dorais and James Hanson using ordinals.

First, assuming weak CH, note that there exists a non-Borel set with cardinality $\aleph_1$. Indeed, take some set $X \subset [0,1]$ such that $|X| = \aleph_1$. The cardinality of the power set $\mathscr P(X)$ is $2^{\aleph_1}$, while the cardinality of the Borel sigma-algebra $\mathfrak B(\mathbb R)$ is $2^{\mathfrak \aleph_0}$, see e.g. here. Hence there exists $A \in \mathscr P(X) \setminus \mathfrak B(\mathbb R)$.

Next one can enumerate $A$ using ordinals as $A = \{x_\alpha\}_{\alpha \in \omega_1}$. For any $\alpha \in \omega_1$ let $A_\alpha := \{x_\beta : \beta \in \alpha\}$. Then $A_\alpha$ is countable and hence Borel. Moreover, for all $\alpha,\beta \in \omega_1$ either $A_\alpha \subset A_\beta$ or $A_\beta \subset A_\alpha$.

For more details about ordinals see e.g. Set theory by T. Jech (2006).