I don't see any unresolvable pin conflicts, so it should be safe to try.
- The GPRS shield uses the UART and some digital pins, but none of the A-marked analog pins.
- The ECG shield uses some of the power pins for power and ADC reference voltages, and A1 thru A6 (selectable with a jumper) for the analog ECG signal back to the Arduino. Either pin D4 or D9 is used for a calibration signal, this is selectable by a jumper. Make sure you select D4, because D9 is used by the GRPS board!
Of course, other factors (such as software conflicts between the Arduino libraries for both shields) could ruin your day. I can't promise it'll work as you intend it to.
Your real challenge will probably be in processing the analog ECG data into something meaningful that can be transmitted over GPRS (determining heart rate, detecting complex events like arrhythmias etc.). You probably shouldn't send the raw ECG waveform over GPRS, as continuous transmission takes a lot of power and will likely yield a large bandwidth bill. Depending on your situation, it would probably be much more efficient to send a digest of interpreted cardiac parameters periodically (and only sending raw waveform data upon request for example).
Hopefully needless to say: do not use this for any health-critical purpose. Neither the Arduino nor the shields and other hardware you're using have been specifically designed or certified for healthcare purposes. Experimenting with ECG equipment is great fun for education and perhaps some sports performance tracking, but you should never consider your project as replacement for seeing a physician and following his/her advice.
Best Answer
It appears that you have a phony HM-10. This is part of the price you pay for ordering cheap from Amazon. If you want to increase range, you can make sure that it's not close to anything metal/conductive.
You could try connecting an external antenna, but you'd have to trim the length to get it right. It's quite likely that the onboard antenna has it's own matching network to match its high impedance to the low impedance of the chip's output. You'll need to fiddle around with wire length to find out how long to be. Be careful, you could destroy the product if you overheat the PCB with your iron, or maybe even blow the chip due to a high SWR.