How to check the speed of a WordPress site on different hosts without WordPress access?

You can’t, and you shouldn’t.

If you did it would not give a true reflection of speed with a full wordpress backend.

Go back to your client and tell them that this isn’t how it works. You’ll need a full copy of the site to test on various hosts.

I’ve been through this many times with clients, and whilst they don’t want you to have the code it’s the only way that you, and they, can ever be sure that the site will be on the “fastest” host.