One could justifiably see spirituality (the Jewish, Christian and Islamic) as ‘Adhyatma’ if one focusses on the psychological transformations that a human being undergoes in the process of moving from the mundane life to a ‘spiritual’ one. You would be closer to the notion of ‘Adhyatma’ if you can also additionally guide such a process. Such guidance requires, of course, that there is some deep understanding of such transformative processes and their mechanisms.
Yet, I would be hesitant to suggest that ‘spirituality’ translates ‘adhyatma’ or the other way round. From among the many reasons, I pick three:
(a) because the weight of the word ‘spirituality’ is drawn from Christianity, i.e., the kind of transformative force on a human being that we have in mind is the work of the ‘holy spirit’, it would be inappropriate to translate ‘adhyatma’ as ‘spirituality’;
(b) it is difficult to characterize the changes in human beings that we are talking about by using this word in Islam and Judaism as well. There is no ‘holy spirit’ in these two religions that does the work of the Trinitarian holy spirit;
(c) since a ‘spiritual’ person is the result of the workings of the holy spirit, in the context of Adhyatma which insists that transformations are the results of human phenomena working on human beings, the use of this word would be inappropriate. Both the Christian and the Adhyatmic use militates against such a translation. In short, no, we should not translate ‘spirituality’ as ‘adhyatma’.