If the watch has watchOS5 then;
This is a known problem ios13 doesn't seem to like older watches with
watchos5 or below,here are some proven solutions from other forum
members.
Quote;
Apple watch 2;
I found a friend, who was on iOS 12 and used his iPhone to pair. Before
pairing it asked to upgrade the watch OS (iOS 13 didn't offer this option).
After I upgraded the watchos I unpaired the watch from the ios12 phone and
then was able to pair it with any iPhone running iOS 13.
Quote;
1st gen apple watch;
pairing was successful on the following device (with associated OS)
to pair via bluetooth I used an iphone 7 with iOS 12, after pairing
immediately accepted the watchOS update to version 4.3.2 after everything
was successfully equipped with the last update, unpaired the iPhone and the
Apple Watch a reset (factory settings) all this took some time but afterwards
pairing on an iPhone XR with iOS13 g was a piece of cake.
Quote;
May 9, 2020 4:58 AM in response to Hashimsabbah
Just did this with a series1. I had to pair with the original phone and update
the firmware for me to pair correctly. All I did was ask to pair and the update
happened on an iPhone 6.
Quote;
nickfit888
Jul 5, 2020 5:23 AM in response to fieryjack
So the “pairing with an old iPhone on iOS 12” trick actually worked for me. I
got a brand new series 3 Apple Watch via a warranty replacement from
Apple. I could not pair the watch to my iPhone X due to the 5.3.7 update
issue. Originally it allowed me to update to 5.3.7 but it was stuck after that.
Based on the advice from “fieryjack” I took an old iPhone 6 (iOS 12.4.7) and
was able to pair my Apple Watch. Once it was paired (watch was fully set
up and synced to the iPhone 6) I tried to update the Watch OS but it said I
needed up update the iOS in order to do so. I unpaired the watch from the
iPhone 6 and paired it with my iPhone X (running iOS 13.5.1). Not sure if
this additional step helped but I chose the manual pair option and this time
the watch gave the option to update to 6.2.6! Once it started updating i
turned off the Bluetooth on my iPhone so that it would update faster using
the wifi network (assuming you are connected to a fast wifi network). I
think it took between 2-3 hours to download the update and after that i was
able to restore my watch using a previous back up.
Thanks again fieryjack for the tip!
Find a phone with ios12 on it if you can and pair your watch to it,it will ask
you to update the watch select yes,after it has updated unpair the watch
from the ios12 phone and then pair it with your ios13 phone.Now you can update to watchos6.
Alternatively contact apple store and see if they will update the watch
through the watch's data/diagnostic port.