The circuit won't work, for two reasons.
First, most gates have limited drive capability. Typical LED requires 20mA, and most gates cannot drive.
Second, the circuits will interfere. Remember that when A or B is low on your circuit, they draw current through their resistors from the other circuit. Consider when your AND gate is on, and A is off. The LED will only see about 1V, which is not enough to drive most LED's.
To solve the first problem, use a CMOS quad NOR with high current capability, like an ACT or AC (both should sink/source 20mA). This gives you rail-to-rail voltage. Buy online from someone like Digikey, Mouser, or Newark in the US, similar suppliers in UK and Australia.
Here's the circuit that fixes the second problem by using the diode to isolate the circuit functions:
A---.|….…….|---.|
B---.|.NOR.--|---.|.NOR.---------|
..…………...…….……....….…|
..…………...…….……....…..LED
..…………...…….……....….…|
….|--.|….…..A---.|…………..…|
B--|--.|.NOR.-----.|.NOR.-- R1--|
..…………...…….……...…..…|
.……...….……………….…...R2
..…………...…….…….........…|
…………..……....................GND
Circuit Operation: There are two parts isolated by the diode. One is above the LED, and one below. The top part powers the LED if A or B, while the bottom part has the current limiting resistor and a dimmer. NAND and NOR gates can be used to make anything. Typically, 4 gates on one chip, so best to use 4 identical gates. I used NOR gates, but good chance 4 NAND gates would work also.
Top Circuit Description:
A and B go into a NOR gate. The output is backwards, so put it into both inputs of a second NOR gate to invert. This goes to the top of the LED.
Bottom of the LED:
B goes into both inputs of third NOR gate. This inverts B. B' and A go into fourth NOR gate. This dimmer signal is on to dim, and goes through R1 to the bottom of the LED. R2 goes from the bottom of the LED to ground.
Sizing the Resistors:
Two constraints are used to size R1 and R2, depending on current you want through LED for bright and dim modes.
Dimmer is zero for bright. Select R1 and R2 so that R1 and R2 together sink the desired current for the LED.
Other constraint is when Dimmer is 1, fourth NOR gate sources some current. Size R1 and R2 for ratio that gives correct current through LED. R1 should probably be larger than R2.
This is way cheaper and easier than gates + relay.
Good luck!