Hi there,
it certainly does affect search rankings. You are changing the HTML, copy, UX - all those changes are important for SEO.
How fast the changes will happen depends on how often does Google crawl your website and look for changes. Also, make sure you update the cache for the new version. After the redesign, it's a good idea to monitor closely the UX metrics, such as bounce rate, exit rate, time on page etc. because those metrics seem to be one of the ranking factors for Google. If those metrics go worse, consider to A/B test some versions of the homepage as well.
But basically, keep the relevant keywords on homepage, include proper internal linking, don't mess with HTML too much and make sure to manage 301's, if needed.
Hope it helps a bit. Cheers,
Martin