No, William can't assume the lease. The lease would be ended early, and I think there's a short period of time left on the period during which Andrew would be entitled to a pro-rated refund on the money he spent on the original renovations.

William would negotiate a new lease, which would probably contain a clause that the lease is terminated if/when William becomes king.

If William moves into Royal Lodge, the place would be protected by taxpayer-funded protection just as it was when Andrew was a working Royal.

Charles would save a bundle by relocating Andrew, and no doubt he tells himself that's the reason why he's playing musical houses.

I don't understand why the tenant, whoever he might be, would have to pay for a new roof. Not sure whether the repairs would be paid for by the Crown Estate or by Charles out of the Sovereign's Grant.

William would probably be allocated an amount for interior renovations, such as kitchen update, but would pay out of his own pocket if the materials he chose exceeded the amount budgeted.