Procress provides a simple way to indicate progress of a process. It was born out of a need to show such progress for AUCTeX processes, but is made general enough to accommodate other processes.

procress can be enabled for AUCTeX buffers as follows (Uncomment the :straight line if you have straight configured, otherwise make sure that process.el is in load-path)

(use-package procress
  ;; :straight (:host github :repo "haji-ali/procress")
  :commands procress-auctex-mode
  :init
  (add-hook 'LaTeX-mode-hook #'procress-auctex-mode)
  :config
  (procress-load-default-svg-images))

Or if you do not have use-package

(require 'procress)
(procress-load-default-svg-images)
(add-hook 'LaTeX-mode-hook #'procress-auctex-mode)

The function procress-load-default-svg-images loads SVG images (copied from Templarian/MaterialDesign) which are shown in the above gif. If your Emacs does not support SVG, or you want to use other images, you would have to configure procress-animation-frames, procress-success-frames and procress-failure-frames. By default these just have basic text.