"Surely it was national park land and not owned by anyone."
National Parks in the UK are not like National Parks in many other countries which are often wilderness areas owned by the government.
National Parks in the UK are simply a planning instrument that controls development both private and commercial. The land will either be owned and farmed by an individual, or be part of a large estate which is farmed by a a number of tenant farmers.
To answer the original question, the New Forest is probably the nearest National Park to London although it is in a highly populated area so you won't be alone in your wandering. You could hike along one of the long distance prehistoric footpaths like the Ridgeway, and just pick a section to walk that fits in with your travel arrangements. The link below is the first one I found for info, I am in no way recommending it or any sort of arrangement.
Walking in the UK is easy but wild walking follows a certain criteria, namely the further north you go the wilder the walking will become. You'll need to go a bit further than you might want to at the moment to get a really wild experience. Snowdonia National Park in North Wales, the oldest National Park the Derbyshire Dales, the Yorkshire Dales, North York Moors or the Lake District, but if you want really wild walking in England then the North Pennines Area of Outstanding Natural Beauty is as wild as it gets. You of course then have Scotland if you have two weeks to spare.
https://www.nationaltrail.co.uk/en_GB/trails/the-ridgeway/Hope this helps.
