Is "no late work" a common policy?
One issue not covered by the many good answers is proportionality.

There's nothing wrong with having firm deadlines and sticking to them, or allowing a grace period with corresponding deductions, and so on. But it's your job as lecturer to make sure the 'punishment fits the crime'. If you have a zero-tolerance late policy and a short time window for a hard assignment that counts for a lot of points, then the policy is disproportionately harsh even if it's fair and clearly stated. I bring this up because you mentioned assignments/projects that carry upto 25% of the grade. 

This is partly why I use a sliding scale late policy, where students can turn in things late, but lose a percentage of their score for each day they're late, upto a week for a 2-week assignment at which point they earn nothing. If you wish to have firm deadlines for a project that accounts for a large portion of the grade, then you might consider creating intermediate deadlines to break up the penalty. This way, students can learn the consequences of missing deadlines without facing huge penalties. It is also more robust to unforeseen events that no one can control.