W5VO is right, the N-MOSFET doesn't turn on because \$V_{GS}\$ is too low; the gate may be at +5V, the source is nowhere near ground. Minimum changes will be to have P-MOSFETs both left and right, and drive one with the other's inverted drive signal. You'll need an extra MOSFET to invert the signal.
clabacchio's solution
is better, but like he says, has a bad static current through the resistors. If the LED voltage is 2V and M2 is on, the node R1/R2 will be at +2V. Current through R2 is then 2V/330\$\Omega\$ = 6mA. Current through R1 is 3V/330\$\Omega\$ = 9mA, so the LED current will only be 3mA, or half the current through R2. For higher LED voltages this gets even worse, and at a LED voltage of 2.5V there won't be any LED current at all.
Decreasing the resistors' values will increase the LED current, but the current through R2 will increase accordingly. This circuit will always have a low efficiency.
\$ \eta = \dfrac{(V_+ - 2 \cdot V_{LED}) \cdot V_{LED}}{(V_+ - V_{LED}) \cdot V_+} \$
That's for LED voltages lower than \$V_+\$/2. For higher LED voltages the LED won't light at all and \$\eta\$ will be zero. For \$V_+\$ = 5V and \$V_{LED}\$ = 2V this gives
\$ \eta = \dfrac{(5V - 2 \cdot 2V) \cdot 2V}{(5V - 2V) \cdot 5V} = 13\% ! \$
This is near the 15% efficiency of OP's circuit.
The solution is to switch off the current through R2 when M2 is on, and that's what an H-bridge does. Replace R2 by another n-MOSFET, and R1 by a P-MOSFET, and place a resistor in series with the LEDs (they can share it, only one is needed).
Of course this will double the number of MOSFETs, and you'll need an extra one to create a complement driving voltage. But it's the only solution to avoid the current losses.
edit (after LED datasheet was added)
The LEDs have rather different forward voltages, so I would suggest to give each LED its own series resistor (not share it, like I first suggested).
Values: we'll ignore the voltage drop across the MOSFETs, so the 660nm will leave 5V - 1.8V = 3.2V for the resistor. At 20mA this gives 160\$\Omega\$.
Same for the 905nm LED (isn't that infrared???): 5V - 1.2V = 3.8V. At 20mA this gives 190\$\Omega\$.