The only way from City of London to ACH is by flight. You should expect to be able to find tickets starting around $353, if you search flights from LHR to FDH. The best time to book long flights is 5-6 weeks before your trip.
For these medium length trips, it's important to book your tickets as soon as possible. While there are flight transportation options available, the sooner to book the better. While prices for buses or trains might not fluctuate as much as flights, their availability certainly will. It's recommended to book these flights at least 2 weeks away, but even more if you are able. Shuttle service may be available to ACH. If you're traveling with a large group, you should probably take a shuttle as it will reduce the price per person. Alternatively, there are shared shuttle services available. With this option, you'll want to book as soon as possible to allow the shuttle service enough time to coordinate your pickup and dropoff.