My bottle brush is looking a little sad and looks like it’s dying. It has a shoot near the ground at the bottom coming out of the
main stem. My bottle brush (Callistemon) is approximately 8 years old and about 5 feet tall and 4 feet spread. It looks like it's dying as the root isn't steady in the ground. Our soil (in Cornwall) is not great and mainly old mining waste. We weren't sure what to do with it, but yesterday I see that a new shoot, approx 3 inches tall, is growing at the bottom. If I can't save the main tree, would I be able to grow a new one from this shoot?
Yes it's possible to water and fertilize to cultivate the new shoot. But be aware that if it originates from the same unstable root system it may become unstable in time as well.
Without a photo it's hard to visualize what you mean by "the root isn't steady in the ground". Or why exactly it is dying. Is the plant wind-blown and uprooted? Are you watering adequately? You may need to work on establishing stability with staking of the plant, soil improvement and water management, before you can hope for any plant to grow well.