Undo the changing of the DNS servers. Put them back where they started.
It is not a requirement to move DNS hosting when building a web site, and if the current provider told you it was required you should cancel service with them.
After putting the DNS servers back to the original DNS hosting provider you only need to update the @ and www records to point the web site traffic to the new web hosting.
If the above is not possible then at the new DNS provider you need to fix the MX record in the zone file.
Ask the original email provider what the MX record should be for the domain you are working on and they should be more than helpful to let you know what it should be to get the email working again.
All of the following are separate services and can be hosted by different providers:
- Who you pay for your domain name (Domain Registrar)
- Who hosts your DNS service (DNS servers / where the zone file is edited)
- Who hosts the web site
- Who hosts the email
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