When the label says "not suitable for outdoor use", it's referring to water resistance. Assuming you're not going to be mowing your lawn in the rain, it'll be fine - just make sure it's suitably rated for the current you're drawing.
Having said that, Ban is right in saying that the best thing is to replace the whole cable. Again, make sure that the flex you buy is the appropriate rating, make sure your connections at the lawnmower end are tight and secure, and if you're using a new plug make sure it has the correct fuse.
Another point, do you have an RCD? Check your consumer unit (fusebox, as they used to be known) - if not, you should be using a plug-in variety.