Prince Andrew has a lease on Windsor Lodge. He (although the Queen may or may not of foot the bill) paid for the renovation that was deemed and necessary and the lease. The lease is, I believe, 75 years, and his heirs can still hold the lease if he dies before the lease is over. So for Harry to move into Royal Lodge, Andrew would have to be bought out.
I really don't think Harry will get the same level of lodging that William & Catherine have been given. The other children of Queen Elizabeth (excluding Charles) have been given "small" apartments (I'm not sure exactly how big of these apartments are, but I am sure they weren't as big as Charles & Diana at Kensington Palace, Charles at Clarence House, or William & Catherine at 1a) in Buckingham Palace as a place to live when they need to be in the city and to work out. Their probably all claim their main residences (and homes) as their country houses. And Anne and Andrew were both given country houses after their marriages, when they needed a home for their family.
I personally believe that Harry won't have a huge city residence, like 1A at Kensington or Clarence House. Maybe if the Cambridges move into Clarence House after Charles' accession, Harry will move into 1A because of the expenses of the renovations and it would seem silly to spend all that money and move out after only a few years.
Remember Harry won't have the same "income" that William will have. Yes, he probably has inherited a bit more money than William would have, but he will probably be living on that plus any inheritance from Charles in the future. He doesn't have future Duchy of Cornwall income, and unless he is supported by his brother or his father (which many people look down on Andrew and Edward for having their mother pay their expenses), he will need his inheritance money to create his income for him.