The escapement on the Oliver typewriter is, for all intents and purposes, just as mechanically simple as the rest of the machine for the most part. It uses a pivot point to turn the downward movement of the universal bar upon a keystroke into an upward movement on the other end which pushes the ribbon vibrator forward and at the same time brings the escapement dog up so that the escapement gear can move one unit upon the reverse operation, when the key is let go of and the system reverts back to its rest position. The second and third photos show this as best as able.
Here is what happens upon the escapement gear and the dog;
Some notes on this subject:
If the spacebar advances the carriage but the keystrokes do not, the issue is thus not with the escapement segment. For the space-bar to work, the whole system above the upside-down T must be working.