I wanted to install Neon on a spare partition on my desktop. My partitioning is:

sda1 - to be used for Neon
sda2 - swap
sda3 - EFI
sda4 - OpenSUSE Leap 15.0

What I did:
Created bootable Neon USB stick;
Booted from stick in UEFI mode
Started installation from running media.
Chose manual partitioning, installed on sda1.
Chose sda3 (the EFI partiition) for the installation of the boot loader.
Continued the installation to completion, removed install media & rebooted.

Grub menu came up as expected.
Selected Neon, got a dark screen with a mouse pointer.
No further progress.
Waited many minutes, needed to do a hardware reset to escape.

On second bootup, selected OpenSUSE, which booted OK.
Rebooted & selected Neon recovery menu. That loaded fine.

So it seems that Grub as installed by Neon is working,

The /neon files on the EFI partition are: - BOOTX64.CSV, grub.cfg, grubx64.efi, mmx64.efi, shimx64.efi
There is also /ubuntu containing fwupx64.efi, grubcfg and /fw (empty)

I've searched for information about dual booting two different linux distros in UEFI, but there was nothing helpful that I could find.

Any suggestions on where I might have gone wrong?

Because You see a mouse-pointer after
booting KDE neon, You're having a problem
with Your "X Window System"-configuration.

Either switch to TextMode-Terminal:
control-alt-F1 or -F2,
login and investigate.

Or:
Boot into TextMode:

Kernel boot-parameter in GrUB-entry:
systemd.unit=multi-user.target

a. login
b. Start: "xinit xterm --"

See, what happens.

The following is displayed on tty1:

[ 1.268556] Couldn't get size 0x800000000000000e
[ 3.552922] sd 4:0:0:0:0: No caching mode page found
[ 3.552950] sd 4:0:0:0:0: Assuming drive cache Write through

xinit xterm on tty2 tells me that "server is already running".

A) Stop the X Window System's server.
$ sudo systemctl isolate multi-user.target

B) Do the "xinit"-thing.
If xterm visible ==> SDDM-problem
If not ==> probably wrong graphics driver

C) If You have network on KDE-neon TextMode:
$ sudo pkcon refresh
$ sudo pkcon update
$ sudo reboot

What file system are you running overall
Are you able to output a log?
Do you have a fsck running on a particularly large drive?
Did installation complete without error?

C) If You have network on KDE-neon TextMode:
$ sudo pkcon refresh
$ sudo pkcon update
$ sudo reboot

This fixed the problem. Thank you.