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).