Previously running 14.1 (lineage-14.1-20180223-nightly-oneplus3-signed.zip)

No need for a full wipe.

Update TWRP

Get latest version from https://twrp.me/oneplus/oneplusthree.html

cp twrp-3.2.1-0-oneplus3.img twrp.img
adb reboot bootloader
fastboot flash recovery twrp.img
fastboot reboot

Make a backup using TWRP

Reboot into recovery, and check that we really have the new version.

Choose "Backup", select all checkboxes and also choose to use compression.

Retrieve the backup and all user data

adb pull /data/media/0/TWRP/backups
adb pull /sdcard

Obtain updated firmware and modem

OnePlus3_5.0.1-FIRMWARE+MODEM-flashable.zip (the 5.0.1 version from here)

There are suggestions elsewhere that without updating these, the rest might fail.

Obtain 15.1

lineage-15.1-20180226-nightly-oneplus3-signed.zip from https://download.lineageos.org/oneplus3

Obtain gapps

As currently recommended on the LineageOS Wiki:

MindTheGapps-8.1.0-arm64-20180223_195845.zip

Install

adb push OnePlus3_5.0.1-FIRMWARE+MODEM-flashable.zip /sdcard
adb push lineage-15.1-20180226-nightly-oneplus3-signed.zip /sdcard
adb push MindTheGapps-8.1.0-arm64-20180223_195845.zip /sdcard

Reboot to recovery. Format system partition. Add the three zip files to the flashing queue (in the order listed above) and flash them all.

Reboot.

Success - the system came back up in a good state.

Problems

So far just that the "back" and "recents" hardware buttons were swapped. There is an easily found setting that can fix this.