$\DeclareMathOperator\GL{GL}$ $\DeclareMathOperator\co{H}$ $\DeclareMathOperator\ko{K}$ $\DeclareMathOperator\trd{tr-deg}$ $\DeclareMathOperator{\ch}{char}$Given a field $F$ and a homological degree $j \geq 1$, define $$\mathbf{N}(F,j) := \min \left\{N : \co_j(\GL_n(F)) \rightarrow \co_j(\GL_{n+1}(F)) \text{ is an iso. whenever } n \geq N \right\}$$ Here the map is induced by the standard embedding $\GL_n(F) \hookrightarrow \GL_{n+1}(F)$ and the homology is with integer coefficients.
(1) It follows from van der Kallen's general result for rings with finite Bass stable range that we always have $\mathbf{N}(F,j) \leq 2j + 1$.
(2) Sprehn–Wahl show in Homological stability for classical groups by reproducing an argument of Quillen that for $F \neq \mathbb{F}_2$ we have $\mathbf{N}(F,j) \leq j + 1$. I believe Quillen's explicit cross-characteristic computations imply if furthermore $F$ is finite that $\mathbf{N}(F,j) \geq \left\lceil\frac{j}{2}\right\rceil$.
(3) It follows from a stable vanishing result of Quillen and the low degree phenomena $\GL_2(\mathbb{F}_2) \cong S_3$, $\GL_4(\mathbb{F}_2) \cong A_8$ that $\mathbf{N}(\mathbb{F}_2,1) = 3$ and $\mathbf{N}(\mathbb{F}_2,2) = 5$. Szymik's computation in The third Milgram-Priddy class lifts yields $\mathbf{N}(\mathbb{F}_2,3) = 7$.
(4) It follows from Lahtinen–Sprehn - Modular characteristic classes for representations over finite fields that for every prime $p$ and $m,a \geq 1$, we have $p^m < \mathbf{N}(\mathbb{F}_{p^a}, a(2p^m - 2p^{m-1} - 1))$.
Edit: Together with the bounds in Randal-Williams's answer below, we get $\mathbf{N}(\mathbb{F}_{p}, 2p - 3) = p+1$ for every prime $p$.
(5) It follows from Theorem 3.25 in Nesterenko–Suslin, Homology of the full linear group over a local ring, and Milnor's K-theory that for infinite $F$ we have $\mathbf{N}(F,j) \leq j$. Moreover there is an exact sequence $$\co_j(\GL_{j-1}(F)) \rightarrow \co_j(\GL_{j}(F)) \rightarrow \ko_j^M(F) \rightarrow 0$$ where $\ko_{\star}^M$ is Milnor K-theory. Thus $\ko_j^M(F) \neq 0$ implies $\mathbf{N}(F,j) = j$.
(6) Springer shows in A remark on the Milnor ring that for uncountable $F$ and $j \geq 2$, the group $\ko_j^M(F)$ is uncountable, and hence by (5) we have $\mathbf{N}(F,j) = j$.
(6') In fact Springer shows the following: let $F_0$ be the prime subfield of $F$ and set $$\delta(F) = \begin{cases} \trd(F/F_0) & \text{if $\ch(F) >0$,} \\ \trd(F/F_0) + 1 & \text{if $\ch(F) = 0$.} \end{cases}$$ where $\trd$ denotes the transcendence degree. Then for every $2 \leq j \leq \delta(F)$ we have $\ko_j^M(F) \otimes_{\mathbb{Z}} \mathbb{Q}\neq 0$, and hence $\mathbf{N}(F,j) = j$ by (5).
(7) Bass–Tate show in The Milnor ring of a global field that for a number field $F$ that embeds in $\mathbb{R}$ and $j \geq 3$, we have $\ko_j^M(F) \neq 0$ and hence $\mathbf{N}(F,j) = j$ by (5).
(8) Friedlander shows in Homological stability for classical groups over finite fields (Theorem 3) that for every prime $p$, we have $\mathbf{N}(\overline{\mathbb{F}_p},j) \leq \left\lceil\frac{j+1}{2}\right\rceil$.
(9) It follows from Theorem 2 in Kroll - The cohomology of the finite general linear group that for every prime $p$ and even $j$, we have $\mathbf{N}(\overline{\mathbb{F}_p},j) \geq j/2$. Together with (8), here $\mathbf{N}(\overline{\mathbb{F}_p},j)$ must be either $j/2$ or $j/2 + 1$.
(10) A result of Galatius–Kupers–Randal-Williams in $E_\infty$-cells and general linear groups of infinite fields (Corollary 9.16) implies that $\mathbf{N}(\overline{\mathbb{Q}},j) \leq \left\lceil\frac{3j + 1}{4}\right\rceil$.
(11, added after skupers comment): It follows from Borel–Yang, The rank conjecture for number fields that for a number field $F$ and $m \geq 1$, the image of $$\co_{2m-1}(\GL_{m-1}(F);\mathbb{Q}) \rightarrow \co_{2m-1}(\GL(F); \mathbb{Q})$$ misses the "primitive elements" in the Hopf algebra sense, which can be identified with $\ko_{2m-1}(F) \otimes_{\mathbb{Z}} \mathbb{Q}$ by the Milnor–Moore structure theorem and the definition of (Quillen's) algebraic K-theory. This latter group is always nonzero by Borel's work, hence for odd $j$ we have $\mathbf{N}(F,j) \geq \frac{j+1}{2}$.
What else is known? Are there general lower bounds for countably infinite fields?
