At the beginning of the story we definitely see Tom behaving almost bratty, and quite childishly. However, throughout many experiences in the story, we certainly can see a change in him which induces one to believe that he must have begun his transformation toward maturity.
While at the start he was misbehaved, towards the middle and end of his adventures we see how Tom starts changing his ways a little bit: He starts getting interested in Becky, he becomes curious about Huck's life of freedom, and begins to see difference between right and wrong.
However, he still retains the basic qualities of childhood such as his fears, his worries, his tendency for mischief, and his adventurous nature. Taking all this into consideration, we could safely assume that Tom was in the transition from childhood to adolescence, and that in the beginning of the story he may have been around 10 or 11 and by the end he is about to become a teenager. The story does not say a specific age.
Prob around 11 or 12