I'm currently working on a homebrew world for a campaign of mine, which is currently on hold indefinitely, so there's no time pressure.

Overall design intent:

1. I don't want my world to be a clone of the general world and lore of D&D (including the Forgotten Realms & the planar cosmology), with the only difference being the Material Plane.


2. I'm not entirely content with the planar cosmology and pantheons in the Forgotten Realms and other existing settings (Eberron, etc).

Since I personally don't like the Astral Plane in particular, I'm thinking about outright removing it from my world.

However, the DMG states on page 43 on "The Planes":

At minimum, most D&D campaigns require these elements:

• [...]


• A way of getting from one plane to another


• A way for spells and monsters that use the Astral Plane and the Ethereal Plane to function

Obviously, I don't have to adhere to these guidelines. However, I'm aware that a number of spells, creatures, magical items and other things in 5e directly refer to the Astral Plane. Miniman's answer to the question What are all the ways a player can get to the Astral Plane? lists quite a few of these.

Naturally, spells (like Astral Projection), creatures (including playable races like Gith) or magical objects that refer to the Astral Plane simply don't exist (in unmodified form) in this campaign. I'd homebrew something for what happens when you put e.g. two Bags of Holding into each other.

I also know that the Astral Plane can be used as a means of travelling between different worlds, using the color pools located on it. I'm thinking about implementing a Yggdrasil-style World Tree in my world, which would assume this job.

Are there any other ramifications as a result of not having an Astral Plane in a 5e campaign setting?

Only the following elements in the core rulebooks use the Astral Plane

While it's impossible to list every possible Astral Plane interaction in D&D, the list of things in the three core rulebooks which rely on the Astral Plane is actually very limited.

The following spells, items or abilities allow travel to the Astral Plane, and will not have that function if the plane is removed:

• An 18th-level monk's Timeless Body ability


• A wild magic sorcerer's wild surge (2% chance)


• The 9th level spell astral projection
  


• 
  Prismatic spray and prismatic wall's violet layer (can also banish to the Ethereal or another plane)


• The Robe of Stars


• A torn Bag of Devouring, Bag of Holding, Heward's Handy Haversack, or Portable Hole, or in some cases one of those placed inside the other


• Ether cyclones on the Ethereal Plane (5% chance)

The only other core rules elements which rely on that plane are as follows:

• Souls of slain individuals traditionally travel to their final resting place via the Astral Plane. You need to invent a new reasoning for how souls get to their afterlife or the realm of their deity.


• The Githyanki live on the Astral Plane. Without the Astral, they either don't exist or live somewhere else.


• 
  Forbiddance blocks planar travel, including specifically from the Astral Plane (as do certain other spells blocking planar travel in general, like antimagic sphere, Mordenkainen's private sanctum and imprisonment); obviously, if you have no Astral Plane, those spells don't do that any more, but of course they still block travel to/from other planes.

You don't need an Astral Plane

Fundamentally, you can completely ignore the Astral Plane. The only significant changes you need to make are explaining how souls and planar visitors get to the realms of the gods, explaining where stuff gets banished by certain spell effects that normally send things to the Astral, resolving very high-level abilities that normally allow astral projection so that they work some other way, and explaining what happens when you place one Bags of Holding inside another.

It's impossible to cover every scenario

You are trying to create a homebrew universe, which is awesome. I wouldn't worry too much about trying to cover all your bases but just deal with issues as they come up.

Building an entire universe is incredibly difficult, so don't worry about covering most cases (because you probably can't.) Worry more about building an engaging world and working on inconsistencies as they come up.

My own world and time management

I'm currently running a campaign in my own contained world that technically doesn't have those planes, either. I haven't even really considered the issues around that and I don't plan to unless something specific comes up. There is so much more to work on that creating contingencies for every possible thing is not a good use of my worldbuilding/storybuilding/encounterbuilding time.

It'll be a process

Authors can spend their lifetimes building and filling out their universe. As GMs, we often don't have that luxury - especially if our players are itching to get started.

What you need to do is fill in enough to get you started. Have your main story arc(s) outlined as well as your primary and motivating characters. But you're going to have to be flexible. Your players are going to change it, and you should let them and listen to them. Let them influence you and help build the world they are playing in. It'll likely lead to greater table satisfaction - and they're going to come up with ideas you probably wouldn't have :) And their actions can introduce new characters you hadn't thought of or new regions/worlds/planes interactions that you didn't consider.