Wow. That's some day trip.
It sort of depends on what you mean by mid-morning. 9 or 10, or, 10 or 11?
2 and a half hours or 3 hours perhaps each way so you and the adult children will have to get a shift on. You won't want a car near Waterloo, unless money doesn't matter for the very high prices for parking, Congestion Charge, ULEZ charge, Sadiq Charge. If that's of no issue then its out the M4 to the Hungerford turn, Jct 14 and cross country.
By train it would be tube or cab to Paddington, the GWR train to Swindon, then bus or cab to the farm.
Their website is many months out of date as far the covid rules and procedures go in England, and they do highlight their Avian Flu problems and procedures.
How did you settle on this day trip? Is it to visit friends or family? If so, could they collect you at Swindon station?