If you want to, you can do both rides in one morning. Here's how:
Be at the security tables 1 hour before park opening (6 am for MM, 7 am for regular day). Send everyone without a bag through security first, then one person remains behind for bag check. Have the people who run through line up at the DL gates. Have everyone line up at a different gate because they are not all open, and they don't tell you for a few minutes which ones will be open and which ones won't.
When the park opens, run/walk as fast as you can to Nemo. You can send one fastpass runner to Space Mountain to pick up fastpasses if you want. They will be able to make it onto Nemo even if you are at the front of the line. Even with 2 small kids, we were on the first Nemo sub leaving at 8:15 am, SM fastpasses in hand.
After Nemo, do whatever you want until 8:45 am. If you did Nemo on a non-MM day, this is immediately after you disembark Nemo.
Go to the Grand Californian entry to DCA. The line for TSMM starts RIGHT IN FRONT of this entrance, so it is a HUGE advantage to be there.
Again, have the people without bags enter before the last person with the bags for security check. They will let you into DCA at 9:30 am. Have the first people entering jump into the TSMM line that is in front of the Grand Californian entrance.
Once the person with the bags joins the TSMM line, have your fastpass runner go to Soarin' and pick up Soaring' fastpasses. Sometimes they give them out at 9:30 am, sometimes not. It's worth checking.
At about 9:45 am the TSMM line moves SLOWLY forward and stops just behind Grizzly River Run. Even if your Soarin' fastpasses person isn't back yet, the line moves slowly and stops again, so they should easily be able to catch up with you.
At 9:55 am, the TSMM line moves again and walks across the bridge to Paradise Pier, and onto the TSMM line. We were the third group in line and were off of TSMM by 10:15 am. Perfect timing for getting on California Screamin'.

After that you should hopefully be able to pick up another set of fastpasses (and also Grizzly River Run fastpasses since it is disconnected) and you can head to whatever other big rides you like (such as Tower of Terror).
We did the Nemo thing once and the TSMM thing twice. It works!