Increase gain in operational amplifiers
The first obvious difference between your circuit and the Sunrom circuit is the driver:



The way the Sunrom drives the transmit coil puts more power into the coil.  That gets you more power out to the RFID tag.  That's more power for the tag to play with, so it will help with range.

125kHz tags actually work by modulating the transmitted signal.  If you transmit a weak signal, you will receive a weak signal.  Look for information on "backscatter" RFIDs.

Next up is the choice of opamps.

Your circuit uses the LM358.

It has a bandwidth of 1MHz at a gain of 1.  At 125kHz, you have a usable gain of maybe 8 to 10.  The first receive amplifier in your circuit is already maxed out - it has a gain of about 10.  You can't change the feedback resistor to try and get more gain - there's none there to be had.

If you want more gain, you need to get a better opamp. With a better opamp, you can change the feedback resistor and get more gain.  You'll want an opamp with much more than 1MHz gain bandwidth product.  There are opamps available with the same pinout as the LM358, but with better performance.  You need:  higher bandwidth, rail to rail operation on input and output, and operation on a single 5V supply.

The maximum usable gain of your opamp is the gain bandwidth product of the opamp divided by 125kHz.  That's where I got the number above for the maximum gain from your LM358.

Last on the list is the antenna. 

The site you got the circuit from describes how to make and tune the antenna. It also explains how the backscatter RFID readers and tags work.

Tuning the coil is not optional.  If the antenna is off, you won't get good range.

Make the antenna as described (120 mm diameter and 58 turns,) and tune as described (use an oscilloscope while adding or removing turns to minimize the glitch or add and remove turns while testing to see what range you get.)

The obvious place to start is the antenna.  Tune it for best performance first.

Then, try using a better opamp and higher gain.  You should be able to do this without making a new board.

Finally, you could redo the output stage for more power output (don't forget to tune the antenna afterwards.)  You can tack it onto you existing board to test, but will need a new board if you get good performance and want to make it permanent.



And now a prize for missing the bleeding obvious:

The original circuit is designed for 12V driving the coil.  You've gone with 5V.  You halved the voltage, which gives a 1/4 of the power output the original had.

Start there.  Proper 12V power for the transmitter.  If that doesn't help, follow all the other steps.