Given that, a digital watch $\text{X}$ beeps every $30$ seconds while watch $\text{Y}$ beeps every $32$ seconds.

Now, digital watch $X$ and $Y$ beep together every $ = \text{LCM}(30,32) = 480$ seconds $ = 8$ minutes.

First, they beeped together at $10\; \text{AM}$. Then the immediate next time that they will beep together $ = 10\; \text{AM} + 8$ minutes $ = 10:08\; \text{AM}.$

So, the correct answer is $(A).$