This circuit should give (regarding resistor values) 0mA at 0V and 20mA at 4,47V. You don't get 0mA, because LT1001 is not single supply Opamp. To get 20mA at 2V, just change combined resistance of R7, R8, R9, R11 to 100Ohm.
That's not true. The circuit is simply cutting lower input voltages because LT1001 isn't suitable for single supply operation. Just use a different OP.