Unfortunately Apple like to play with themselves and no one else is allowed in the sand pit.
The apple routers all work well together and for doing music etc the Airport express works very well with the TC.. I don't know if that is your Time Machine router but I am guessing so.
To wirelessly connect the WET610N needs to be in bridge or universal repeater mode.
You need to set the TC to short wireless name, no spaces using only alphanumeric characters.
Also lock the channels.. as wireless bridging is very sensitive to channel swapping in normal auto setup.
I really doubt a firmware upgrade is relevant. But if you need to do it.. plug the WET straight into a computer with ethernet and work from there. Normally these devices have no dhcp set, so you will need to statically set the IP.. i cannot see an OS change doing anything to stop a normal http upgrade but you might need to use firefox or chrome rather than safari.
Or find a nice PC user.. friend.. they are not all scum of the earth.. and ask for help.