If you are hoping to have a quick and painless house move, here are two things you must do to achieve this.
Create a moving-day schedule
Rather than simply agreeing to a time when the removalists can show up and then leaving the rest of the day to chance, you should create a moving-day schedule that describes exactly what you want to get done by each hour of the day. Giving yourself mini-deadlines to accomplish specific tasks will give you the push you need to avoid procrastinating and ensure that you don't dawdle to the point where you cause a delay.
For example, if you're planning to dismantle a few pieces of furniture by 7 am, just before the removalists are expected to arrive, putting this in your schedule (and possibly setting an alarm to encourage you to wake up and get started as early as possible) will increase the chance of you finishing this off by the time the removalists need to put the dismantled items in their van. This, in turn, should mean that they won't be delayed as a result of waiting around for you to complete this job.
Additionally, having this schedule will ensure that you don't forget about specific important tasks that you must get done on this day. This might include, for example, taking cash out of the ATM to give as a tip to the removalists or draining your jacuzzi or swimming pool before you lock up the old property.
Give the removalists detailed information about your belongings ahead of time
The other thing you must do to make your house move easy is to give the removalists detailed information about your possessions before they come and take them away. For example, you might want to tell them which pieces of furniture you want them to dismantle for you (and send them photos of this furniture) and let them know if you are taking any of the carpets with you (as you'll probably need their help to roll these up). Additionally, you might want to provide them with a copy of your new home's floor plan that specifies which items go into which rooms.
This will mean that when they get to your current house, they can start dismantling specific items or aiding you with the removal of certain carpets straight away, without first having to have an information-gathering conversation with you. Likewise, giving them the floor plan and instructions on where to put certain boxes will mean that they won't have to stand around and wait for you to show them which boxes go where; instead, they can get on with completing this final stage of the move as soon as they arrive.