I would recommend a Typhoid Mary (carrier) sort of situation, if it was me. This is likely best done as a multiple dispersal vector virus, or a single dispersal vector but not as a direct bite situation. Especially as you want the original infection to be introduced via the water supply, and water can carry viruses rather easily:
Viruses are a major cause of human waterborne and water-related diseases.
A percentage of the population carries the zombie infection without turning. They 'escape', or are 'rescued', either by dumb luck or they don't 'smell' human to the turned zombies, making them not interesting as food. Alternately, they are clear of the origin zone before anyone realizes there's a zombie infection, so they have no idea they're infected with something and spreading it. Either way, they reach thus far uninfected portions of the population and disperse the infection like any other virus: surface contact, interpersonal contact, bodily fluids, sex, tears, sneezing, whatever. The uninfected don't realize the carrier is a carrier until it's too late.
If you're rooting for the zombies, there's no clear indication from blood or genetic material or whatever whether someone is a carrier or purely uninfected, and only discernible evidence of infection is that occurring during/after the turn. Also, shooting the zombies, and even setting them on fire, increases the risk of blood-borne infection spraying everywhere, or in the case of fire, not wholly burned particles containing the virus becoming airborne. This could potentially cause a bigger problem than it would solve. Breathing zombie smoke or blood spray, or even skin absorption, makes this war even trickier.
Bear in mind that zombies aren't necessarily dead, though this is a popular choice. Alternately, a colony of zombie virus might animate a person or possibly an animal by combining with the individual cells, if it reproduces rapidly enough and gets a good hold. There's also various parasites and forms of venom which can cause mind control effects in their targets, and the zombie virus could act similarly:
Zombies may still be a thing of fiction, but some parasites more or less turn their hosts into the walking dead. These masters of mind control manipulate their hosts from within, causing them to act in self-destructive ways that ultimately benefit the parasite.
This isn't to say that these aren't stereotypical zombies, only that the methodology of zombification isn't necessarily death, but easily some form of biological manipulation of one kind or another.
In other words:
What are the physical requirements of the zombies in order for them to not be immediately killed off?
These would be the biological requirements, such as ability to be detected before turning, ability to transmit without being infected (carriers) rather than the purely physical (strength, speed, intelligence, etc.)
What would the human-to-zombie timetable need to look be (Approximately)?
This is a complicated issue, but I'd say probably anywhere from 6-48 hours, with carriers. Possibly determined by different people's biological and/or genetic makeup. Differences in infection time would make it more complex to pinpoint the carriers or have a distinct idea how long one has.
Approximately how many initial zombies (town's population) would it take to infect the country with those attributes?
I'd call it around 10% of the infected being carriers, too few and fewer would spread to other locations, too many and the carriers' existence would be noticed much more quickly.
Given the current tech level of Earth, is it feasible for zombies to take over even with their physicality?
With carriers, it could be, if the carriers did not show indications and neither did the infected until the turn itself began, however long that took. Bite marks are clear indications, so to increase the odds of the zombies outnumbering (I hesitate to say taking over, since they really just want food, right?) the uninfected the infection vector would have to be more viral in nature than violent, though of course we would also expect the turned to start eating people just because it's what they do.
This is my first answer submitted on Stack Exchange, I hope I did it correctly and that it may provide you with some useful ideas.