I cannot receive emails to firstname.lastname@example.org or email@example.com
Migrating to the new cPanel means updating the DNS as well. However, according to who.is, you have no MX records present for your domain.
Please go to cPanel > Set web address > roadtyresuk.com > Manage > Modify MX records, and configure your website to use Gmail as provider.
Or else, you can create an e-mail forward, which will forward all received e-mails to a custom e-mail address. Go to cPanel > Manage Email > Create to use this feature