The idea is pretty cool, but it would need a lot of work to really shine. I can't code/mod for shit, so all I have is idea input.
That said, this mode could actually be AMAZING with the right things, and its a shame you're deving it with the idea of no downloads. When I hear "Zombie TF2", I think of things like Til Death Do Us Part (and Part Two) which paint a good picture of what a gamemode zombie TF2 could be like. Dark levels, waves of enemies and and a feeling of hopelessness. The idea of permadeath helps with the "hopelessness" concept, but would really need a dark map to highlight that, as a fully sunny map like upward doesn't really bring out the "zombie apocalypse" feel. Another thing to take from the videos (and the zombie genre in general) is the idea of infection. One idea you MAY be able to use with your no downloads standards, would be the death of one of the four players becoming a zombie itself. A super zombie even, using the "undead" halloween skin for their class. It could be player or CPU controlled, but for the "hopeless" factor CPU may be better, with the former player getting a third person view of what their former body is now doing to their team.
tl;dr See first sentence.
EDIT: Another thing I just realized. You said you wanted this to be no download so anyone can go and play it...but with a four player limit, whats the point? You wouldn't be able to load this in a public server with 24/32 people. What are you going to do, put them in spec? They'd just leave. Plus, without the more heavy polish, four friends aren't likely to make a small server just to play it. It seems the idea behind it clashes with the reality.