The basic idea of gradient descent is:
Calculate the gradient of some score with respect to parameters that you can control
Take a step in the direction of that gradient that improves the score (subract a multiple of gradient - for gradient descent - if you want to minimise some cost function)
The backpropagation using chain rule in neural network layers is part of the first part, calculating the gradient.
If you interleaved these steps during a single calculation, by taking an update step before propagating the gradient over all layers, you would not propagate the true gradient, but some interim value. There is no guarantee this value would reflect the true gradient of the affected layer when compared to training data and loss functions.
There are some manipulations of the gradient that are acceptable and improve convergence. For instance momentum, and various forms of weighting based on previous gradient calculations. However, I am not aware of any successful attempt to perform updates with partially-calculated gradient then attempt to continue the interrupted gradient calculations over the mixed updated and not-yet-updated parameters.
It is also possible to calculate gradients and then only update some of the possible parameters. In some cases this is useful, for instance it is a good choice when performing transfer learning, by only updating a few layers close to the output. This constrains the network to keep a lot of its trained parameters as-is, which reduces the chances of over-fitting to the smaller dataset that the network is learning from when transferring.