You should not get mild flu symptoms at all from the flu jab - there is no live flu virus in it so it is impossible to get the flu from the jab.
The only side effect from the flu is possibly tenderness and swelling at the site of the injection.
It takes 12 days for the vaccine to become effective, so you can develop flu within this time span. If you caught flu just before you had the vaccine you will still get the flu.
However, the flu jab cannot guard against all the strains of the flu, and the World Health Organisation has to determine which strains to vaccine against some months before the jab is available.
So yes, you may get flu after having the jab, but you are less likely to.
The flu jab does not prevent colds and many people mistakenly refer to colds as the flu