The ESP is a 3.3v device. If your Uno is a 5v device, its A/D converter can measure and report voltages from 0 to 4.9878v (5v minus 1 LSB). If the same signal is presented to the ESP, any value above 3.29919 volts (3.3v - 1 LSB) will read as 4095, i.e., out of range.
(If this isn't your issue, adding wiring diagrams of each system to your question will help us understand it better.)
Perhaps a different ESP unit would be the ideal solution?
As far as I know, all of the ESP's are 3.3v devices. So the solutions I see are:
Power your sensor with 3.3v (is this an option?) so its output will be within the input range of the ESP's built-in A/D.
Scale the sensor output from it's 0-5v range to the 0-3.3v range the ESP can accept. Whether you can use a simple divider network or would need an op-amp may depend on the sensor, but if, as I suspect, it is a simple resistance device, a voltage divider should work.
Use an external 5v A/D converter. This will require a 5v power supply (which I presume you are using already, to power the sensor) and voltage level conversion between the A/D and input pin(s) of the ESP. The conversion could be a purchased level-converter board or a simple resistor-divider network.