Years ago I remember trying to play WC4 with a mouse and found it required a joystick. Is this the case with this version? And is privateer the same?

Yes, you can play both games with mouse. However, joystick is recommended (at least for WC3&4) as it is more enjoyable this way.

I played both WC4 and Privateer with a mouse and keyboard and it's absolutely playable. Still the WC series was never easy to control, no matter what controller you use. You have to get used to whatever you choose.

It's better to play Privateer using mouse because of some Dosbox problem with joystick handling (control isn't precise). AFAIK there is special (modified) version which solved this problem. I'm playing Priv with mouse because of control and lack of space for joystick on my desk.