Short version: this site has no accounts, no forms and no comment boxes, so it never receives your name, email or payment details.
What the hosting records
Like any web server, the hosting provider keeps technical logs of requests: time, page address, country, browser type and an IP address. They are used to keep the site running and to count visits in aggregate.
What a button click records
A click on a brand button passes through a redirect on this domain. It stores the page you were on, which button it was, and your country and region. It does not store your IP address in that record, and it sets no cookie.
Cookies and trackers
The site sets no cookies of its own and loads no advertising pixels or social widgets. The one outside script is Cloudflare Web Analytics, which the host adds to each page: it loads from static.cloudflareinsights.com and reports page views and load times back to this domain. Cloudflare states that it does not collect or use visitors' personal data. Fonts and images are served from this domain.
After you leave
Once you are on the casino's site, its own privacy policy applies. Read it before you register; the casino will ask for identity documents.
Questions
Write to tui@mummys-goldcasino-nz.com.