Irretrievable breakdown of marriage is the only ground for a divorce i the UK - adultry, violence and so on are only the causes of the irretrievable breakdown. Separation is another cause of the breakdown.
Anyway, that is beside the point. You can't be expected to live under the same roof if relations are bad - it's not good for the child. If you can't decide between yourselves the courts will get involved and they have several choices.
The well being of the child is always the first consideration, and if your wife can afford to live in the house with the child whilst you live elsewhere (with your financial support), that is best as there is less disruption. Usually the condition is the house is sold when the child is 18 and the proceeds of the sale divided.
If that is impossible the court will order the house be sold and the wife will get a larger share of the equity as she has to provide a home for your child, assuming there is equity.
No court will order your wife and child out of the family home without provision, whilst you remain there.
With a three year old child it is only in very exceptional circumstances the father is given sole residency.
If you have any sense at all you will both come to an amicable agreement and involve the lawyers as little as possible.