I honestly don’t know, because it is definitely caused by either a 3rd party plugin or a custom code – the issue is unfortunately completely unrelated to both Elementor and Ajax Search Pro. If you deactivated Ajax Search Pro and tried the pagination, it is still broken because of the permalink structure.
These issues are very rare and are usually caused by a custom code snippet. If you recently added a code to change the search page permalink structure, that can cause this problem.