The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in particular:

$\mathbb{F}_1$ seems to have a unique extension of degree $m$, generally written $\mathbb{F}_{1^m}$ (even if this is slightly silly), which is thought to be in some sense generated by the $m$-th roots of unity. See, e.g., here and here and the references therein.

If $G$ is a (split, =Chevalley) semisimple linear algebraic group, then $G$ is in fact "defined over $\mathbb{F}_1$", and $G(\mathbb{F}_1)$ should be the Weyl group $\mathcal{W}(G)$ of $G$. For example, $\mathit{SL}_n(\mathbb{F}_1)$ should be the symmetric group $\mathfrak{S}_n$. (See also this recent question.) I think this is, in fact, the sort of analogy which led Tits to suggest the idea of a field with one element in the first place.

These two ideas taken together suggest the following question:

What would be the points of a semisimple (or even reductive) linear algebraic group $G$ over the degree $m$ extension $\mathbb{F}_{1^m}$ of $\mathbb{F}_1$?

My intuition is that $\mathit{GL}_n(\mathbb{F}_{1^m})$ should be the generalized symmetric group $\mu_m\wr\mathfrak{S}_n$ (consisting of generalized permutation matrices whose nonzero entries are in the cyclic group $\mu_m$ of $m$-th roots of unity); and of course, the adjoint $\mathit{PGL}_n(\mathbb{F}_{1^m})$ should be the quotient by the central (diagonal) $\mu_m$; what $\mathit{SL}_n(\mathbb{F}_{1^m})$ should be is already less clear to me (maybe generalized permutation matrices of determinant $\pm1$ when $m$ is odd, and $+1$ when $m$ is even? or do we ignore the signature of the permutation altogether?). But certainly, the answer for general $m$ (contrary to $m=1$) will depend on whether $G$ is adjoint or simply connected (or somewhere in between).

I also expect the order of $G(\mathbb{F}_{1^m})$ to be $m^r$ times the order of $G(\mathbb{F}_1) = \mathcal{W}(G)$, where $r$ is the rank. And there should certainly be natural arrows $G(\mathbb{F}_{1^m}) \to G(\mathbb{F}_{1^{m'}})$ when $m|m'$. (Perhaps the conjugacy classes of the inductive limit can be described using some sort of Kac coordinates?)

Anyway, since the question is rather speculative, I think I should provide guidelines on what I consider an answer should satisfy:

The answer need not follow from a general theory of $\mathbb{F}_1$. On the other hand, it should be generally compatible with the various bits of intuition outlined above (or else argue why they're wrong).

More importantly, the answer should be "uniform" in $G$: that is, $G(\mathbb{F}_{1^m})$ should be constructed from some combinatorial data representing $G$ (root system, Chevalley basis…), not on a case-by-case basis.

(An even wilder question would be if we can give meaning to ${^2}A_n(\mathbb{F}_{1^m})$ and ${^2}D_n(\mathbb{F}_{1^m})$ and ${^2}E_6(\mathbb{F}_{1^m})$ when $m$ is even, and ${^3}D_4(\mathbb{F}_{1^m})$ when $3|m$.)