I have a Weebly site and I wan't a free custom domain email. There is an option of G Suite that's like $4/m but I found a Migadu which is free. The thing is that Migadu wants me to change the name servers of my website which I can, but I don't know how it will affect the site. Will I lose anything?
Your site will go down if you change the Name Servers.
Ask Migadu if they have any MX records that you can apply to the domain instead. That is really all that is needed to transfer the email hosting to another provider. If they don't, you may want to seek out another provider.
Migadu hosts only email, so you can leave your site with your current web hosting provider or move to another. Being an email hoster, they should only be asking you to change your email servers. Your DNS servers are maintained by your website hosting provider. You will have 2 providers not 1. Migadu will do all the email, so it will read email@example.com. And your web hosting provider will handle all traffic to and from your web site mydomain.xxx.
Migadu should ask you to change your mail servers: SMTP, IPOP (if applicable), incoming/outgoing. And of course you will need to change any mailto's on your site to reflect your new mail servers (mailto:firstname.lastname@example.org).
Your domain name servers (DNS) should not change because that routes traffic to your website itself. Those need to point to your web hosting provider.
Check this link: https://hc.weebly.com/hc/en-us/articles/201740093-How-Can-I-Change-My-Site-s-Address-or-Domain-Name- The instruction might be helpful. You can avoid this problem with WordPress or Mobirise for example. You can change the domain in the app very easy.