Will removing the turf in the garden cause drainage/flooding problems?
Our garden is all grass and we are planning on cutting all the turf out this month and then landscaping the garden a section at a time over the winter.
Will removing the grass cause any drainage/flooding problems when its rains as it will all be mud for a while in the garden?

Only if you had a drainage problem already. But you will walk in mud. I would mulch it up for the winter so weeds don’t start growing.
What causes flooding is when people remove turf and concrete the yard so the rain has nowhere to soak in. However you may find it better and less messy to remove just the amount of turf you need to for each for each section as you come to it. It will avoid the ground getting muddy and unmanageable. Also if we have a mild winter weeds will probably start to grow in the bare soil so by the time you come to the last sections you will still have to dig it.
If you have the energy it is better to bury the turf as you cut it, which will improve soil texture and fertility and avoid the problem of getting rid of it. You remove the first line of turf and put it to one side. Then dig out the soil and barrow it to the end of the section you are going to do. Then remove the turf from the second row and put it upside down into the first trench. Cover it with the soil from under the turf you’ve just removed, which will make a second trench etc.
When you come to the last trench, fill it with the soil from the first one. This is the traditional way of starting a garden. It is hard work but it keeps you fit and gives you a better result.