F-Secure Key is a capable password manager with clients for Windows, Mac, iOS and Android.
After the initial installation you're invited to create a new Key account, which is as simple as entering a password. You're then able to import your passwords, at least in theory (the program can only import XML files exported by other password managers, including KeepPass and Password Safe).
If you can't access your existing passwords then it's easy enough to enter them manually. Provide the login URL, your user name and password (there's a good secure password generator, if you need one), and repeat the process for as many sites as you like.
There are several ways to enter these passwords at some later point. You can copy them to the clipboard, for instance. Click in the password field and a button appears; click this, choose the current site from the dropdown list, and the password will be entered. Or you can have both the user and password entered automatically on Android, Windows and Mac (select the user field, press Ctrl+F2, choose the site; on Android, tap the user name field, switch to F-Secure Key's virtual keyboard and select your login details).
While this all works for free, syncing passwords across all your devices requires purchasing a subscription (as we write, it's around $2.43 a month, $20 a year; LastPass provides unlimited mobile app use with syncing for an annual $12). Once you've set this up, clicking the "Generate Code" button on one device creates a sync code, and entering this on another synchronises the two. Repeat as often as necessary.
There shouldn't be any need to worry about the security of your synchronised passwords, either: F-Secure says they're AES-encrypted, the company never sees any user's information, and can't identify which data belongs to a particular user anyway. What's more, the Key servers are "owned and operated by F-Secure within the European Union", so getting them to hand over whatever logs they have won't be easy.