Yes, but, I would give some thought to what we're doing and why we're doing it. There's a bit of circular logic here and it runs into conflicting objectives.
In theory, you could rewrite history with a 51% attack against your own chain and that would possibly be relatively easy in a private chain scenario where you control a majority of the miners.
The question that arises is this. If your goal is to reserve for yourself the possible of rewriting history, why use an append-only blockchain? Why not use a database with replicas and overwrite capability? The latter is well-solved and would probably be less trouble. Reserving this re-write capability would bring the authenticity of the private chain into doubt in any case in my opinion .
In other words, in my opinion it's not possible to engineer things such that a rewrite of history is probable and also retain the unique benefits of an immutable blockchain. These are conflicting objectives.
Maybe consider a contract design that can amend stored values in a forward-only auditable way.
Hope it helps.
p.s. It would happen as fast as you can splice a new block at the fork point and your miners a new longest chain.