Is it permissible to use the words "the First," "the Last," and "the One" in certain colloquial contexts to denote time, order, or the speaker, or is that considered an encroachment upon the Beautiful Names of Allah?

The names mentioned (the First, the Last, the One) are among the shared Beautiful Names of Allah, which are applied to Him, Glorified be He, and to others. They are not exclusive to Him, the Almighty. There is no harm in applying them in a relative context. However, Allah's being the First and the Last is absolute, as stated in the Hadith: "O Allah, You are the First, so there is nothing before You; and You are the Last, so there is nothing after You."