If I apply 1W to these series-connected didoes ...
We don't talk about "applying power" to a diode. We pass current through it and power is dissipated in the process. Power dissipated in a device is given by \$ P = VI \$ where P is in watts (W), V is the voltage across the device in volts (V) and I is the current through the device in amps (A).
The voltage across a typical silicon diode is about 0.6 to 0.7 V over a wide range of currents.
If we pass 1 A through your series connected diodes they will both dissipate \$ 0.6 \times 1 = 0.6 \ \mathrm W \$.
The 1 W diode will survive.
The 0.1 W diode will rapidly heat up and the internal junction will be destroyed. The diode will fail.