The Setup

To achieve what you want, you'll simply have to wire all the light circuits in parallel off the common of the 3-way. Basically instead of a light on the common, you'll have light circuits. Something like this.

Following this scheme, each single pole switch will control the lights in a single area. While the whole circuit, is controlled by the 3-way switches. This way, if the 3-ways are OFF all the lights will be off. If the 3-ways are ON, the individual single pole switches will determine which lights are on.

The down side

While this configuration is possible, it may be confusing to any future owners of the home. This may also cause problems if the area is ever finished and actually divided into separate spaces, since a somebody could potentially be lost in the dark.

Code compliance

I'm not sure at this point whether or not this setup is code compliant, but a quick call to your local government should yield the answer. I will continue to review the National Electrical Code (NEC), to determine if there are any sections that would not allow a setup like this. If I find any restrictions, I will update this section of the answer.

_{If any others happen to know any codes that might apply here, please post the code section as a comment.}

We start by numbering the cables in your diagram from left to right:

1. is an always-hot feed backstabbed into the single pole switch (it's always hot because it's backstabbed at the same end of the switch as the incoming hot feed).
2. is another always-hot feed (were I to guess, this is the circuit coming in, but there's an off chance some dunderhead wired it to come in on cable #1 instead). I can tell because of how it's double-looped at the switch terminal screw then pigtailed off to the rest of the switches.
3. is the outbound switched hot from the single pole switch to whatever lighting is on it. (It is connected to the other end of the switch from the hot terminal.)
4. is the outbound dimmed hot to its corresponding lighting load.
5. and 6. are the curious cases -- you'd expect both travelers to be in 6. but perhaps the two cables are run together as part of the same circuit in a California 3-way configuration? (The traveler terminals on a 3-way switch are both at the same end.) Either way, they're irrelevant to your timer trouble.

From here, we can tell where your error is -- you wired the switched hot on the timer to cable #1's hot, when the original single pole switch was switching cable #3's hot.

So, turn off the power to the circuit, then unnut the hot (black) and switched (red) wires on the timer, and unwind the switched wire from cable #1's hot and cable #3's hot from the pigtail junction. Then use new wirenuts to nut cable #1's hot into the pigtail junction and cable #3's hot to the switched wire from the timer. Put everything back together, turn it back on, and enjoy your new timer!