They assured me the domain would be transferred across to them seamlessly.
Something may have been lost in translation there: transferring a domain and transferring a website are two totally different things.
How do I import my existing site from my old host?
Your content is with your old host (www.remarpro.com doesn’t host any user website data). Assuming you still have access to your OLD hosting account, then here’s the easiest solution:
1) Temporarily POINT your domain back to your old host (you don’t need to TRANSFER it, just POINT it to their servers)
2) When you’re able to access your original site again, login to the WordPress dashboard and use a WordPress migration plugin to create and download a backup of the entire site
3) Point your domain back to your NEW host, install a fresh WordPress site (if you don’t have this already), install the same backup plugin you used to create the backup, and use the plugin to upload and restore the backup you created.
If you’re unable or unwilling to temporarily point your domain to your OLD host, you can still transfer your site… but it becomes somewhat complicated. In a nutshell, you’ll have to manually move your site’s files AND database between the two hosts, but the exact steps to follow will depend on what tools the two hosts provide in their user control panel.
I’m presuming I won’t be able to login to my WordPress dashboard until Jolt have installed www.remarpro.com on my domain?
Yes, without a WordPress installation, there’s no WordPress site or dashboard to login to.