Can you reboot your system, not just a scanner/printer?
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Almost afraid to, but I'll get back to you on the other side.
Did you press Return key to finish installation after this?
Still not detecting the scanner.
I'm sure I hit enter after the process ran.
I just re-ran it and got this:
hockey_steve@ThinkCentre:~/Downloads$ sudo dpkg -i *deb
[sudo] password for hockey_steve:
(Reading database ... 289152 files and directories currently installed.)
Preparing to unpack brscan4-0.4.10-1.amd64.deb ...
Unpacking brscan4 (0.4.10-1) over (0.4.10-1) ...
Setting up brscan4 (0.4.10-1) ...
This software is based in part on the work of the Independent JPEG Group.
hockey_steve@ThinkCentre:~/Downloads$ sudo apt install gdebi
Reading package lists... Done
Building dependency tree
Reading state information... Done
gdebi is already the newest version (0.9.5.7+nmu3).
0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded
Please download this file.
https://support.brother.com/g/b/downloadend.aspx?c=us&lang=en&prod=mfcj5620dw_us_eu&os=128&dlid=dlf105200_000&flang=4&type3=566
Then double click on it to install with gdebi.
brscan4-0.4.10-1.amd64.deb
Don't know if it counts.. but is it really necessary to add a printer to the system by using Settings - Printers .... because what if you don't, won't it show?
That's why I recommended to install from the terminal with installer script at the beginning. This script will install printer/scanner/fax all in one go.
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Can you download this driver and install it with gdebi?
https://support.brother.com/g/b/downloadend.aspx?c=us&lang=en&prod=mfcj5620dw_us_eu&os=128&dlid=dlf101194_000&flang=4&type3=559
Done
Still no scanner detected
The printer was added when I did the re-install of Zorin.
I wonder if I should delete the printer and the drivers and run the package install
I think there is no deed to remove printer driver.
But you could reboot and reinitialize your printer to see if it makes any difference.
I've rebooted the computer and the printer several times already.
What worries me is that you cannot uncompress the installer file in the File manager.
You will have a problem in a daily operation if you cannot uncompress file.
Can you open File Roller and try to uncompress file from there?
Or try uncompressing online from a website.
I suggest you to chage the name of the installer something simple such as
brother (it will still run I tested it myself)
Open terminal
cd Downloads
sudo bash brother
CUI will ask you
Input model name ->
Type the printer name exactly as follows:
MFC-J5620DW
Follow the instruction thereafter.
We're getting somewhere.
But I still can't scan after rebooting both computer and printer.
The printer is connected by WiFi, the PC by LAN
How are you connecting the printer?
USB or WiFi?
If it is WiFi then you need to connect the printer to your local network.
Assuming your printer got a similar control panel as mine, go to set up then press WiFi assistance (mine is in French, English wording may be different).
With Win10 the scanner worked using WiFi to the same network connected to the PC by a LAN connection.
Guess I'll have to run out and buy a LAN cable.
I'll let you all know if that solves the problem
In the meantime, thank you for all of the assistance.
Cheers 
You cannot connect printer with LAN cable to the computer.
You need to connect it to the router.
Is there anyreason you do not want to WiFi connection in Linux?