There are ways to do this, but the method depends on several things.
In many cases, you can access the files simply by browsing through Windows Explorer(not Internet Explorer) which should show all non-hidden files. If encryption was used there are ways to access it.
Remember, as now an external drive that I assume is not configured to be "bootable" you will not have a login screen for that drive. The info is still there, but the method of access is different.
You have to change the permissions on each drive if you're going to use hard drives across systems. The "access denied" occurs because your laptop user has access to the folders, but your user on your other computer doesn't.
You need to boot from the laptop hard disk, right click on the folder, look for properties and then security. You'll want to give the group everyone full control. Then you can reboot from your other hard drive and give your user permission. Then take away full control to everyone and manage them from there.
Hopefully you have a way to plug your laptop drive back into your laptop.