Here are the first, more straightforward, differentials in the AHSS $$H^p(X;KO^q(\ast)) \Rightarrow KO^{p+q}(X)$$ for real K-theory. Note $KO^q(\ast)$ is $$ \begin{cases} \Bbb Z &\text{if }q = 8k,\\ \Bbb Z/2 &\text{if }q = 8k-1,\\ \Bbb Z/2 &\text{if }q = 8k-2,\\ \Bbb Z &\text{if }q = 8k-4,\\ 0 &\text{otherwise.} \end{cases} $$

• For $q=8k$ the map $$d_2: H^\ast(X;\Bbb Z) \to H^{\ast+2}(X;\Bbb Z/2)$$ is $Sq^2 \circ r_2$, where $r_2$ is reduction from integer cohomology to mod-2 cohomology.

• For $q=8k-1$ the map $$d_2: H^\ast(X;\Bbb Z/2) \to H^{\ast+2}(X;\Bbb Z/2)$$ is $Sq^2$. (These two differentials are what Denis Nardin alluded to in the comments.)

• For $q=8k-2$ the map $$d_3: H^\ast(X;\Bbb Z/2) \to H^{\ast+3}(X;\Bbb Z)$$ is $\beta_2 \circ Sq^2$, where $\beta_2$ is the Bockstein from mod-2 cohomology to integer cohomology.

• For $q=8k-4$ the map $$d_5: H^\ast(X;\Bbb Z) \to H^{\ast+5}(X;\Bbb Z)$$ is $\beta_2 \circ Sq^4 \circ r_2 + \beta_3 \circ P^1 \circ r_3$, where $P^1$ is a Steenrod operation on mod-3 cohomology. (Or perhaps this is $-d_5$, depending on choice of generators.)

The only remaining $d_3$ differential, from $E_3^{p,8k}$ to $E_3^{p+3,8k-2}$, is given by a map $$ \Phi_{2,2r}: \ker(Sq^2 \circ r_2) \to \mathrm{coker}(Sq^2). $$ This is a so-called secondary cohomology operation, associated to the relation $Sq^2 \circ (Sq^2 r_2) = 0$ between cohomology operations. There is no straightforward description of $\Phi_{2,2r}$ in terms of primary operations. In the same way that $Sq^1$ in $H^*(X)$ detects the presence of degree-2 attaching maps $S^n \to S^n$ among the cells of $X$, and $Sq^2$ detects the presence of the Hopf map $\eta$ in an attaching map $S^{n+1} \to S^n$, the operation $\Phi_{2,2r}$ detects the presence of $\eta \circ \eta$ in an attaching map $S^{n+2} \to S^n$.

I am not certain of the twisted K-theory versions of some of these, but this is a deficiency in my knowledge of real twisted K-theory.

Here are the first, more straightforward, differentials in the AHSS $$H^p(X;KO^q(\ast)) \Rightarrow KO^{p+q}(X)$$ for real K-theory. Note $KO^q(\ast)$ is $$ \begin{cases} \Bbb Z &\text{if }q = 8k,\\ \Bbb Z/2 &\text{if }q = 8k-1,\\ \Bbb Z/2 &\text{if }q = 8k-2,\\ \Bbb Z &\text{if }q = 8k-4,\\ 0 &\text{otherwise.} \end{cases} $$

• For $q=8k$ the map $$d_2: H^\ast(X;\Bbb Z) \to H^{\ast+2}(X;\Bbb Z/2)$$ is $Sq^2 \circ r_2$, where $r_2$ is reduction from integer cohomology to mod-2 cohomology.

• For $q=8k-1$ the map $$d_2: H^\ast(X;\Bbb Z/2) \to H^{\ast+2}(X;\Bbb Z/2)$$ is $Sq^2$. (These two differentials are what Denis Nardin alluded to in the comments.)

• For $q=8k-2$ the map $$d_3: H^\ast(X;\Bbb Z/2) \to H^{\ast+3}(X;\Bbb Z)$$ is $\beta_2 \circ Sq^2$, where $\beta_2$ is the Bockstein from mod-2 cohomology to integer cohomology.

• For $q=8k-4$ the map $$d_5: H^\ast(X;\Bbb Z) \to H^{\ast+5}(X;\Bbb Z)$$ is $\beta_2 \circ Sq^4 \circ r_2 + \beta_3 \circ P^1 \circ r_3$, where $P^1$ is a Steenrod operation on mod-3 cohomology. (Or perhaps this is $-d_5$, depending on choice of generators.)

The only remaining $d_3$ differential, from $E_3^{p,8k}$ to $E_3^{p+3,8k-2}$, is given by a map $$ \Phi_{1,1} \circ r_2: \ker(Sq^2 \circ r_2) \to \mathrm{coker}(Sq^2). $$ This operation $\Phi_{1,1}$ is a so-called secondary cohomology operation, associated to the relation $Sq^2 Sq^2 + Sq^3 Sq^1 = 0$ between cohomology operations. There is no straightforward description of $\Phi_{1,1}$ in terms of primary operations. In the same way that $Sq^1$ in $H^*(X)$ detects the presence of degree-2 attaching maps $S^n \to S^n$ among the cells of $X$, and $Sq^2$ detects the presence of the Hopf map $\eta$ in an attaching map $S^{n+1} \to S^n$, the operation $\Phi_{1,1}$ detects the presence of $\eta \circ \eta$ in an attaching map $S^{n+2} \to S^n$.

I am not certain of the twisted K-theory versions of some of these, but this is a deficiency in my knowledge of real twisted K-theory.