Bac Lava: recommend swimming around high tide (the river is tidal as far as Pulborough - four hours after Littlehampton).
High tide here is one and a half hours after Littlehampton. An beautiful, remote location. Gorgeous views over the river and the Sussex Downs on the way to a "dead end" - a few houses and a church - and access to the river (across the bridge).
From Arundel, past the castle public entrance and the Wetlands Centre (well signposted), left fork up behind the "Black Rabbit" (on river bank and good food and drink for after a swim), then left at T junction. Parking a similar situation as at Bury. Swim from upstream or downstream from over the bridge.
There are points to bear in mind for both. The Arun has the fastest tidal flow of any river in England. Swimming is much influenced by the state of the tide. There are thus several possible "styles" for a swim. 1. Go around the top of the tide for a leisurely swim from/around/back to the entry point. 2. Go, say, an hour before high tide and swim with the flow upstream and back as the tide turns. 3. Do a distance. Say, Bury to just north of Arundel (or vice-versa) and have a non-swimming partner meet you at the destination with your clothes
You are likely to be alone, housing distant, with only the occasional canoeist or boatie. (in summer, much of the bank is shielded by reeds - "skinny" is a possibility).