I understand that the Heathrow Express goes to Terminal 5 but, from that terminus, I'd need to transfer to Terminal 3 for my flight.
From the terminus at T 5, how will I transfer to T3? And, how much time should I allow to arrive at the check in area in T3?

Assuming that it is a landside transfer get the Heathrow Exp from T5 to Heathrow Central. It is free. Get on the back on the train then the nearest exit wil take you closest to T3. I think the Heathrow Exp is every 15 minutes but they go to T4 as well on a seperate Branch so I am not sure if they are every 15 minutes from T5, if you see what I mean.
According to this they are 4 an hour from T5 to Heahrow Central.
Doesn't paste very well
From
Then minutes past the hour
Until
Monday to Saturday Departures
Every 15 mins from 5:12 - 23:48
Journey time: 21 minutes
05:07 05:27 05:42
57 - 12 - 27 - 42
23:53*
Sunday Departures
From 5:03 - 08:48
Average journey time: 21 minutes
05:03 05:18 05:48
03 - 18 - 33 - 48
08:48
Sunday Departures
From 09:00 - 21:27
Average journey time: 21 minutes
09:00 09:13
12 - 27 - 42 - 57
21:27
Sunday Departures
From 21:48 - 23:48
Average journey time: 21 minutes
21:48
03 - 18 - 33 - 48
23:48

Are you trying to get fron central London to terminal 3? The Heathrow Express also stops at Terminals 1 and 3 before going on to the terminus at Terminal 5.

sorry, I was unclear in my original post. I'm trying to get from Paddington to terminal 3, and t appeared that the Heathrow Express stopped at T5, so I was trying to figure out a simple way to get from that stop at T5 to check in for my flight, which will depart from T3..
does the Heathrow Express also stop at T3?
Have you considered the cheaper alternative Heathrow Connect of the even cheaper Piccadilly Line?

Info at https://www.heathrowexpress.com/timetable-schedule/london-heathrow
If you’re going to Terminal 1 or Terminal 3, board any train and get off at Heathrow Central. Journey time is 15 minutes.
For Terminal 5, get on the London-Heathrow train at Paddington and stay on board at Heathrow Central – all trains continue to Terminal 5, arriving six minutes later.
For Terminal 4, change at Heathrow Central and catch the free transfer service. These connecting trains depart every 15 minutes and travel time is four minutes.