Trevally knows of which he speaks
However it sounds to me like the ships you're sinking aren't registering with the mission - if the 'progress bar' you get when you click on it out at sea isn't filling up, then you're sinking ships outside the mission area, and they won't count towards the tonnage total. You'll have to get closer to the red line along the British coastline and find ships there. Once you start seeing the bar fill up after a sinking you know you're close enough - just carry on until the bar is full.
Once the bar is full the mission will remain on the map, but will no longer be selectable when in port, and you'll get the campaign star. Then you can move on to the next one.
Trevally modestly didn't mention it, but one of the best mods at the moment (definately a 'must have', along with IRAI, FX Update & New UIs with TDC) is his & Zedi's 'Open Horizons II' - a fully redesigned campaign with much better objectives. If (when, lol) you start to get annoyed with the stock campaign, that mod should be your first port of call. And if you get lost in the mods (or our suggestions don't solve your problem), just ask.
