I can't say how easy it would be to use another company's processor in the AxeFX. I do know the Helix is powered by a Freescale DSP which is supposed to be a beast, so I suppose there are alternatives.
I'm no expert in embedded systems and have very superficial knowledge of these things myself, but my understanding (which may be totally wrong) is this.
1. TigerSHARC was the most powerful DSP available. Maybe something from Texas Instruments was also as powerful, I'm not sure, but the DSPs used in other products, including Helix, are weaker.
2. Axe FX relies on a lot of horsepower both for modeling and effects. The amp models are very complex, all those knobs cost something in terms of CPU cycles, and effects are very CPU consuming as well. You can hear the difference between Axe and Helix reverbs even on YouTube demos if you pay attention.
3. You can't just fit Axe FX into something considerably weaker. Maybe some things can be optimized but a lot of what's going on inside is pure math, it needs power. So making an Axe with that Freescale DSP is probably not an option.
4. The whole super high power DSP thing seems to be dead now, nobody is developing anything. Sales were rather poor, these chips haven't been updated for ages, production lines no longer exist or are being dismantled.
5. For some reason increasing the number of DSPs is problematic as it increases latency. Maybe there are solutions, I'm not sure.
6. So it may not be a matter of switching to another company's alternative chip, but changing the architecture quite dramatically. It seems that users of high power DSPs are switching to these new FPGAs, which are totally different beasts as far as programming is concerned.
7. Axe FX was a weird device in that the cost of the DSPs was outrageous to begin with. Nobody else does it this way, AFAIK. I think those FPGAs aren't cheap(er). And development costs are higher.
So, from my layman's perspective Cliff and Co. have one hell of a choice to make. And it will be interesting to see what comes out of it for us users.