Here's how I would do it... your mileage may vary; yada, yada, yada...
WEEK 1 - About 15-30 minutes per day
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I would start with "Collect". Don't worry about processing for now. Just start a list -- get yourself a new notebook and start listing your commitments. All of them. Anything that you have to (or want to) do, just write it down.
You can do this in short bursts -- while you're waiting for your food at a restaurant, while you are waiting for others to arrive to a meeting, during commercial breaks during your favorite TV show, while you wait for a bowl of soup to heat in the microwave, etc...
Don't worry about capturing duplicates; that will happen. Better to capture something twice, than to miss it altogether.
WEEK 2 - about an hour on day 1, then 30 minutes per day thereafter
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If you have been writing down everything that has your attention for the past week, you should have about 90% of your commitments captured -- by this point, you may have even completed some of them. Cross the completed stuff off your list.
Now, take the list and label it "Someday/Maybe". You have just created your first GTD list.
Next, pick the most important/urgent/critical things from the list... the ones that MUST be acted upon this week. Write these on a new list called "Projects", and remove them from the "Someday/Maybe" list.
For the remainder of the week focus mostly/only on the items in your "Projects" list. Anything new that comes in, write it on a separate sheet of paper labeled "Inbox", and at the end of the day, decide for each item on the Inbox list, "Do I need to focus on this during the remainder of the week?" If so, write it on the "Projects" list. If not, write it on the "Someday/Maybe" list. If it is not actionable at all, then file it away as reference material. Then, throw away the "Inbox" list -- you will create a new one tomorrow.
Review the "Projects" list every morning, and formulate an action plan for the day. Review again at noon, and adjust your plan as necessary. Then just before you leave for the day, review it one more time and formulate a plan for the next day. If there are certain items on the "Projects" list that somehow elusive, not really moving forward despite your best efforts, then apply the "Next Action" technique as described in the book.
WEEK 3
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Repeat WEEK 2
This is a very light GTD implementation. It does not address some of the more powerful concepts (the most powerful, in my view, being the use of Next Actions and Contexts), but it should help manage the chaos at least a little bit. And it will make the initial days of a "proper" GTD implementation a little easier since you will have already captured a significant percentage of your "open loops".
In fact, the simple act of just writing down all of your commitments may give you enough of a feeling of control that you might actually find time to do a full-blown implementation sooner than you had anticipated.
I hope this helps.