You can't 'simply log on' anywhere and buy shares. You need to open a brokerage account which is a similar procedure to opening a bank account...proof of address/identity etc.
You have to open a stock broking account - online is cheapest and I know ODL Securities in London tick alot of boxes for Irish-based investors and offer a flat fee of €30 in all Irish shares quoted on the London Stock Exchange not matter what size of trade done. I'm aware that TD Waterhouse are similar.