Any Python developers on here? Or UX developers?

I'd remove the spaces before you break down the postcode.

If I put in 'A B123XY'
then the breakdown isn't right.

But it's trivial to fix someone putting a space in the wrong place since every postcode has a space before the last 3 characters, regardless of whether it's 5, 6 or 7 characters long.

As to the order of statements, you would put the most restrictive first.
Or you could check everyone and set a flag somewhere.

But I'd avoid putting every combination of error messages in.
You've only got 3 things to check for.
Too long, too short, bad characters. So it's manageable to put the combinations in (e.g. too long and bad characters). But consider if you had a dozen things to check for. You'd have a screen full of elifs and nested crap. Easier to check one thing at a time and potentially just have two error messages.
"Too long"
"Bad characters"

isValid = true
if length < 6: isValid = false
if length > 8: isValid = false
if set('[~!@#$%^&*()_+{}":;\']+$').intersec­tion(postcode): isValid = false

if isValid == false: print "something's wrong with the postcode"

etc.

Can simplify the stuff where you're working out the parts of the postcode too

print('This tool breaks down UK postcodes to their component parts')

exit=False
while not(exit):
    postcode=input('Please enter your postcode (0 to exit): ')
    postcode=str(postcode).upper().replace(" ", "")
    valid_postcode = True
    
    if str(postcode) == '0':
        print('Goodbye')
        exit = True
    
    else:
        if set('[~!@#$%^&*()_+{}":;\']+$').intersection(postcode):
            print('Your postcode contains an illegal special character, please try again')
            valid_postcode = False

        if len(postcode) < 5:
            print('Your postcode is too short, please try again')
            valid_postcode = False

        
        if len(postcode) > 7:
            print('Your postcode is too long, please try again')
            valid_postcode = False
        
        outward_code = postcode[:-3]
        inward_code = postcode[-3:]
            
        postcode_area=outward_code[:-2]
        postcode_district=outward_code[-2:]
        
        postcode_sector=inward_code[:1]
        postcode_unit=inward_code[1:]
        
        if valid_postcode:
            print('')
            print(outward_code+' '+inward_code+' breaks down to:')
            print('Outward Code: '+outward_code)
            print('Postcode Area: '+postcode_area)
            print('Postcode District: '+postcode_district)
            print('Inward Code: '+inward_code)
            print('Postcode Sector: '+postcode_sector)
            print('Postcode Unit: '+postcode_unit)
            print('')

(caveat - I probably have just as many bad habits as you, I've only been messing about in Python for a few weeks too)