Sequential processing of uncorrelated measurements in Kalman Filter
I think it is evident in both cases. It follows directly from
$$ P(A|B,C,D) = P(((A|B)|C)|D)$$
.i.e.
$$P(x[k]\!=\!\hat{x}[k]|\!\hat{z}_1[k]\!=\!z_1[k],\!\hat{z}_2\![k]\!=\!z_2[k])\!=\! P\Bigg(
  \Big(
      x[k]\!=\!\hat{x}[k]|\!\hat{z}_1[k]\!=\!z_1[k]
  \Big)|\!\hat{z}_2[k]\!=\!z_2[k]
\Bigg)$$
As long as you don't do a prediction update it should be fine.
For nonlinear system, you won't get the same exact estimate, because you will end up with a new dynamics, input, measurement and feedback matrices on every "single-measurement" update