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Head out towards Arundel? You can use the reverse of this route: https://cycle.travel/map/journey/49948
It worked pretty well for me and was a nice route, lots of lanes and quiet roads.
The route from Arundel to London is also pretty scenic until you get closer to town: https://cycle.travel/map/journey/49947 to be honest, I'd probably hop on the train at somewhere like Tadworth.
After ideas for a short ride from Portsmouth in any direction. Considering to ride out in the afternoon, wild camp, ride a bit next day and catch a train back to London.
Going to explore the obvious choice of venturing into South Downs first. Cheers