Is your current account mortgage a tracker rate, i.e. the interest rate at a fixed margin over the ECB base rate? If so, I'd be reluctant to give that up, as the rate you'll pay after your fixed rate expires is highly unlikely to be as good as the rate you're paying now.
As far as I know, you cannot fix on a current account mortgage without switching to a "non current account" mortgage. So you may incur legal and valuation fees for doing so.
Also - if your current account mortgage is the same as mine, any overpayments you make can be withdrawn if needed, e.g. in the event of being out of work for a while.
Personally, I'd hang on to your current account mortgage. It can also be a good place to house your cash savings as you'd be getting an effective rate of interest of 2.25% on them, which is the same as 3% interest elsewhere as you'd pay 25% DIRT tax elsewhere. Not the best deposit rate in the country at present (see Best Buys on Askaboutmoney) but not bad and helps you towards your goal of paying off the mortgage by the time your oldest child starts college.