You don’t have to spend $1000 but with limited mobility, maybe a really nice hotel with views would be something to consider, as well as amenities if touring becomes too difficult.
Unfortunately to see the architecture in these cities, it’s hard to avoid walking and especially on uneven, stone surfaces, not to mention some places in London around the City which could be uphill.
I agree with the suggestion to use taxis or Uber instead of going underground up and down stairs for the subway. Rome has very limited metro lines.
You can figure out bus lines with Citymapper app but would need mobile data.
Just walking along the Thames and Seine will let you see the grand architecture in those cities. In Rome, the most scenic sights are Piazza Navona, Campo dei Fiori and Spanish Steps (its possible to get up there by taxi so you don’t have to climb the steps). Of course Vatican and St. Peter’s Square for more grand architecture.
For nature, you can take day trips out of the cities, though London and Paris both have large parks, expansive green spaces, such as Hyde Park, St. James Park in London and Jardin de Luxembourg in Paris. There is also Champ de Mars, by the Eiffel Tower and Jardin de Tuileries, which runs alon the Right Bank of the Seine between the Louvre and Place de la Concorde.
For day trips, I haven’t been to Cambridge and Oxford but much more bucolic. Other popular places would be Windsor Castle, Stonehenge and Bath. I took a coach tour which went to these 3places.
I don’t know if people go to Brighton on a day trip but it’s a seaside town which seems to be relatively close to London.
For Paris, you have Versailles, with huge grounds and garden. Faux-le-Vicomte is another palace with gardens but on a much smaller and grander scale. IIRC Louis the 14th confiscated this palace from one of his ministers and then decided to build Versailles as well. Giverny is suppose to be a lush garden which inspired Monet in his later years.
From Rome, there is Villa d’Este. Some operators will take you as far as Florence and bring you back in the same day but that sounds exhausting. You may be able to go to Orvieto on a day trip. Won’t see big Renaissance buildings there as you would in Florence but Orvieto is up on a hill overlooking the surrounding landscape.