Do you need a PLL? What are your distortion requirements, linearity of FM deviation versus input voltage? Don't rush off down the PLL route without having first looked at direct modulation.
Start with the specifications you require for the FM signal. Linearity and noise, as well as modulation bandwidth and deviation.
It is far simpler to modulate a VCO directly, lower power, if you can tolerate any tuning law non-linearity. Different VCOs have different linearity. A wide deviation VCO can be used in a small, nearly linear, part of its tuning curve.
Even if you've tried direct modulation and found it wanting, don't give up on it without a bit of investigation. I personally have used a VCO which had a kink in its tuning bandwidth curve at about 500kHz. The main LC tank tuning went via inductors, but the gain/sustaining tuning went via an RC. The manufacturer was persuaded to modify it to use inductors both both tuning points, which increased the flat bandwidth to more than 10MHz.
A colleague of mine made a TV transmitter exciter which started with two VCOs being modulated differentially and mixed, to get a low frequency, wide deviation, high linearity, low noise, wide bandwidth FM signal.
If you do end up using a PLL, then 10MHz bandwidth is fairly straightforward. You could use a digital phase detector. They don't quote a bandwidth, because it's as wide as permitted by the input rates, and limited by what you do with it afterwards. However, for wide bandwidths like 10MHz, I'd tend to use an analogue phase detector, an RF mixer with a DC-coupled IF. It's no faster, but is quieter and better behaved.
The CLTF can be unstable, it depends on the ratio of time constants. It's easier than you think to make a stable loop, and with minimal mathematics, and no Laplace functions.
First of all, short circuit any integrator capacitors in your loop filter, and open circuit any lowpass filter capacitors, so it merely becomes a gain of R2/R1. Now compute the gain round the loop. The frequency at which it becomes 0dB is your loop bandwidth. If you want a different loop bandwidth, alter the gain.
Note that the loop filter time constant plays no part in the selection of the loop bandwidth. It is solely the loop gain that sets this.
Now while the PLL you've created like this has the correct loop bandwidth, and is stable, it probably doesn't meet your specifications yet.
To improve low frequency tracking, put C back in series with R2, to increase the gain at low frequencies. This configuration is called a 'broken integrator'. Keep the C.R2 break frequency no higher than half your loop bandwidth. This will ensure that the phase shift it creates is small enough at your loop bandwidth frequency to maintain loop stability. If there's any tendency of your loop to vary the gain, perhaps with a VCXO with a non-constant tuning sensitivity, then this will vary the loop bandwidth, and may encroach on your low frequency break point, making the loop less stable, and ultimately unstable. Move the breakpoint down further if this is the case. Move it down to at most one third of your loop bandwidth if you're going to use high frequency filtering as well (next paragraph).
If you want to improve reference rejection, you can add a capacitor in parallel with R2, to roll off the loop response at high frequency. Keep the this break frequency at least two times the loop bandwidth if used by itself, or at least 3 times if used in conjunction with a broken integrator.
It might be worth revisiting your specification of 10Hz for loop bandwidth. If that's based on reference rejection, you will often get better loop dynamics with a wider loop bandwidth, with the addition of some high frequency roll-off to improve the removal of the reference modulation.
If you're familiar with a Bode plot, then sketch out what I've described.
Best Answer
No, you've misunderstood how a PLL demodulator works. The NCO (or VCO in an analog system) must track the instantaneous frequency of the incoming signal, not just its average value. If it does this accurately, then then the control value is an accurate replica of the original modulating signal.
If the oscillator only tracks the average frequency, then you have a PM (phase modulation) demodulator, which is related, but different. With wideband FM (such as used for FM broadcast), the instantaneouis phase error will be much greater than 360° at times, and the output of your phase detector will "wrap", createing a lot of distortion.