"quicker turn around"
…. what?
shares that go up quickly?
shares go up, shares go down (something like buses) there will be another dip along soon if you miss this one.
What bank are you with?
Most banks will have have some form of share dealing, even in your branch you will be able to buy shares and get a share certificate (old style). But setting up an on-line share-dealing account will take time.
TD Waterhouse seem quite popular.
"Something i could put £500 into this week but cash in any time from the next day to 3-4 months in the future without going though all the hassle/paperwork of an ISA and using my ISA allowance."
3-4 months is a short time in shares if you are hoping to make money on £500.
If it was £1million that you were investing then that is different.
It might be a year before you make a profit in shares, or break even:
http://www.bbc.co.uk/iplayer/money_box