Can I replace an AND gate with two diodes?
Yes, should work in your limited situation...where inputs are driven from HCMOS logic. This "diode AND" gate's output should drive a HCMOS input. An additional limitation would not allow two or more of these gates in series, without a HCMOS buffer between them.
Noise immunity is compromised by this gate, because a logic low at the output does not fall close to zero volts, but hovers one diode-drop higher. If you use common silicon diodes (1N914), the diode adds about 0.64v to the logic zero. If your HCMOS logic runs with a low supply voltage like 3v, then the logic threshold voltage might be around 1.5v, leaving you with about a volt for noise immunity. A HCMOS AND gate has better noise immunity.
Current consumption of this AND gate is high, and dependent on logic state, compared with an equivalent HCMOS AND gate. To make this gate fast-acting, the pull-up resistor must be small (1K ohm or less). If either input is logic "low", current draw from the Vdd supply, through the driving gate to ground is large. This single gate may draw more current than your entire CPU. Be aware that this pull-up resistor must go to the same Vdd supply that powers the rest of your HCMOS CPU.
Diode speed shouldn't be a problem. Schottky diodes would be preferable, with high-speed silicon diodes a second choice. Even Germanium point-contact diodes could be used.