If am booting Ubuntu from a flash drive on a window 7 OS will any downloads be saved on flash drive, or will they be sent to resources of the windows based machine.

You will need to clarify what mean by "on a windows OS". Are you running Ubuntu in virtual software installed on the windows OS?
Do you have a Live installation of Ubuntu you are booting from?
Do you have a Live Ubuntu on the flash drive with persistence?
Did you do a full install of Ubuntu to a flash drive?

A live system won't save anything on reboot and it won't write to windows unless you take specific steps to do so.

If you are booting to Ubuntu you can save things to any disk it can access.
There is no such thing as a Windows or a Linux machine as such. It is the current active operating system that is important.
It is true that certain OSs use there own preferred file system and file structure, but that applies to all operating systems and most OSs can use most file systems.

furthermore you can also specify the target directory of your downloads, so it depends on you.

With an install system or live CD or USB, everything fits initially within memory. On such a system, when you save something, it saves it to the filesystem in memory. Memory contents are lost on shut down and/or reboot.

But you can specifically tell the live system, or the install system, to save a file to a real drive. Then it will be there after reboot. I used to have a system that I always booted from a CD. The hard drive just held user data and set up scripts to tweak the system, so I wouldn't have to do it over and over each time I booted

In addition to the replies above, if you're booting to Ubuntu from a flash drive and if you have mounted your Win 7 drive to Ubuntu while it's running.

You can save files to Win 7 drive and it won't be lost when you shutdown your Ubuntu system, just make sure that Win 7 drive is mounted. Of course, you need to know the folder where you save your files.