Just spent 90% of my Sunday getting my inbox to zero. No time for weekly review, although try to keep reviewing on the fly. Lack of time at work to make progress on lists due to meetings is pushing things over to my days off. And with no ability to back out of said meetings, I'm not sure how to break the cycle.
I've been in the exact same place man.. My solution so far has been to do a scan of all my new inputs and process everything that is somewhat urgent... then take all other 'stuff', whatever format it is in and put it in a folder called BACKLOG. Use your system as best as possible so that you are continually building the habit of processing new inputs, making next actions and engaging with your system so that nothing sticks. If you dont keep building the habit doing those things, then your system will stagnate and then you will just end up with that huge backlog again.
Then when you get time free, process that BACKLOG folder. And keep doing a weekly review, so that you tie everything together and keep your working system fresh. As TesTeq mentioned above, do the weekly review, of current, actionable (and urgent) things first, then work on getting your inbox/backlog to zero
Maybe someone has got a better solution than that. But I had the exact same thing and this is the best solution that I could come up with.. Would love to get someone elses feedback on that!