Just uploaded new site, but old info on google

When I googled the new site I just uploaded, under the title of my site in the search results, it still has all the text about the website reconstruction.

Will this eventually switch over or is there something I’ll need to do?

It will sort itself out in time. Googlebot will keep coming back to your site every few days (or more!) to check that everything is still there and as it thought - it will soon notice that the site has changed, and shortly after that will update its index with the new information.

you can upload a sitemap to google webmaster and check how many of your most recent urls are already indexed. There might be a lag in the information provided, but it is quite useful.

yes it will eventually show your updated results as googlebot takes some time to crwal your new index pages.

Okay so I just told my client it’s something that happens and will take a short while to correct itself.

use google webmasters tools and submit ur site and sitemap. After a few days it will get sorted out as googlebot revisits a few time and do not find the old items.
P.S.: Increase the crawl rate to minimum time interval for a few months. So that it happens faster

Probably, Google has not index your new website. Submit the URL & Sitemap to go Google Webmaster Tool.