My itinerary on Friday, Sept. 20th begins in Littlehampton with a 10 AM visit to the Sir Harold Hillier Garden, then an 11 AM stop at Furzey Gardens, followed by a lunchtime stop in the New Forest at the Knightwood Oak. Then I plan to visit Bournemouth Lower Gardens, followed by a stop at the Cerne Abbas Giant viewpoint, and then spend the night in Weymouth. What driving delays or difficulties can I expect on this travel day (parking in Bournemouth, for example)?
Another day of, I am sorry to say, an unrealistic timescale. Littlehampton to the Hillier gardens at that time of day could take 1.5 hours. More if there are any traffic incidents.
It will take half an hour to drive from the Hillier garden to Furzey, so that would only allow you half an hour to park, get tickets and see the Hillier garden.
Traffic delays are common in the UK. Parking can also be an issue. Finding your destination can also be an issue so have a good satnav.
You are trying to cover too much in to short a space of time in a congested area.
In Bournemouth, I would suggest you visit Compton Acres Gardens - which is on the west side of the town - really a part of Poole. I would also suggest that you visit the spectacular coast at Lulworth Cove & Durdle Dor which lies at the centre of this map:> https://www.google.co.uk/maps/search/knightwood+oak/@50.6550755,-2.1881213,28444m/data=!3m1!1e3!5m1!1e1
Forget about the Giant on this day as you will have no time.
https://www.comptonacres.co.uk
https://www.lulworth.com/visit/places-to-visit/durdle-door/