WorldPay says they are unable to do this without cancelling the Future Pay Agreement.
World Pay has no functionality to defer payments. So there's nothing that can be done about this. They'd have to cancel the subscription and renew it when they can. You can offer them a coupon to discount their next renewal to free if you like. If World Pay let you pause subscriptions you'd be able to pause it then edit their subscription in CBSubs > Subscriptions by clicking the # under the Id column and extending their expiration date. As far as I'm aware only PayPal and Stripe have pause functionality.
This is fine but we need the customer to sign up to the new agreement BEFORE we cancel the old agreement, but not have the new agreement to start taking payments until a set date.
For this usecase I'd make a new plan just for this scenario. It can be a copy of your previous plan. You'd then set the initial price to $0 and the duration to however long you want to defer payments (e.g. 1 month, 2 months, 1 year, etc..) then of course have your normal regular price and duration same as the other plan. That should work for your needs if I understand correctly.
I can't find on your subscription system where we can get them to sign up but defer the first payment until a specific date. I can see we can make the first payment 0. However, that only gives one month grace period.
Initial payment can have whatever duration you like so if you want it to be 2 months set the initial duration to 2 months for example.