As for video: if your watermark isn't big there's a rather simple solution - using any video cropping tool (e.g. VidCrop) you can cut off the part of your video containing this watermark
that or you can use an app to edit each frame adding a transparrent blurr/smudge effect over the watermark
Sometimes if it was very mild semi transparent channel logo for example this can work reasonably well... but will still be noticable
Sorry I can't remember any specific app name - was a Long time ago I played with something like that.
But am sure many app could do this - kinda like applying your "blurr/smudge" effect as your logo... this is not a image added rather than a photoshop effect applied to each frame of the whole video in the same spot.