Arduino Leonardo, like the Uno, uses an NCP1117 linear regulator (LDO), which can supply 800 mA, but at 11.1 V input that would be 4.88 W, so thermal protection will set in much earlier.
Using USB is a possible solution, but keep in mind that USB doesn't have to supply more than 100 mA, you'll have to negotiate for 500 mA. And Arduino is not designed for low-power, I don't know what the Leonardo consumes, but I guesstimate the Uno's supply current at 50 mA, so powered from USB that may leave too little for for instance a series of LEDs.
Your own switching power supply looks like the best solution. Bypass the NCP1117 and supply it directly with 5 V. A 12 V to 5 V buck converter should be able to have a near 90 % efficiency. If the dingus would need 50 mA it can run for more than 48 hours on the LiPo battery.
Further information
Arduino Leonardo schematic
My advice here would be to try to get a 5 V power supply with at least 1.5 A output just to be sure. Then get a linear 3.3 V regulator. A heatsink may come in handy as well, but most likely won't be needed if my assumptions are correct.
Here's my reasoning: Mbed uses 3.3 V I/O, so it would be best to run digital circuits at same voltage as to not have any potential problems with voltage translation.
So run everything except the Mbed and the E.C. circuit from the linear 3.3 V regulator.
The temp probe needs to be sampled with ADC and you'll get a bit extra resolution if you can run it from 3.3 V, since each bit of the ADC will represent smaller value. Also if the Mbed is going to sample it, then it would make process a bit easier due to reference voltage being more or less same.
The pH sensor is rated to run fine with 3.3 V operation, so just to make sure that there are no problems related to serial port high/low levels, run it from 3.3 V.
The WiFi is 3.3 V already, so no need to comment much about that. I guess that you're going to try to power it from Mbed. If that's the case, then it's good since it shouldn't interfere much with operation of other components.
The Mbed itself has its own 3.3 V regulator, so it would go directly to 5 V power supply.
Finally we have the E.C. circuit, which to me looks the most problematic. It's 5 V and Mbed is 3.3 V, but has 5 V tolerant I/O. If the circuit will work fine with 3.3 logic levels for input, then that's good. Otherwise some level translation may be needed. Unfortunately, the datasheet doesn't say anything. Next we have the "as close to 5 volts as possible" and "ripple and noise free" requirements which feel a bit unprofessional to me, since it isn't really explained what happens when voltage isn't exactly 5 V and there's no such thing as ripple and noise free. For that reason, I'd add a low ESR capacitor as close to its power supply pins as possible and maybe even a ferrite bead on the 5 V line, just to make sure the noise from power supply is away.
Best Answer
No, this is not a good idea.
Check the datasheet of the regulators, but most don't like reverse voltage.
Parallel voltage regulators can't be counted on to share current. One will always have a little higher setpoint than the other. This could also lead to instability, depending on the nature of the controllers in the regulators.
A better way to solve this is to combine the two voltage sources before a single regulator. At this low voltage, you can use Schottky diodes. Put one diode in series with each voltage source. Power will then automatically be taken from the higher of the two voltage sources. Make sure to put something like a ceramic cap after the diodes physically close to the input of the regulator.
You can still use multiple regulators to spread the dissipation and to reduce voltage drops to distant parts of the circuit. You bus around the higher voltage out of the diodes, then regulate that locally as needed. However, you don't tie the outputs of multiple regulators together. You have each power a different part of the circuit instead.
If you want to minimize local dissipation, you use a buck switcher after the diodes. This makes a little more than the minimum input voltage of the regulators. You filter that a little and bus that around. Then you make the regulated voltage from that as needed locally. For example, if using a 5 V LDO that requires 5.5 V in, you might bus around 6 V. Each local regulator would then be 83% efficient.