I've been fairly unimpressed with HSBC in Belfast - customer service is ropy, and EUR account is too much hassle (they won't sort out the tax for you - pay it to UK, you must reclaim it and pay DIRT here), you only get telephone not internet banking, and you get charged €5 pm.
So saying, they have premier accounts you might be eligible for - only in Sterling though: for these, they will exempt you from the tax, give you a free account, give you better telephone banking, and also internet banking. Problem is finding out all this - the account opening lines know nothing, and you can't call Belfast directly - so you must go up even to get the info. You have to make the appointment through the phone line, but when you get there ask them about the Premier accounts if you're interested.
An alternative for you might be Barclays Wealth - you choose your account location: most are offshore, but if you choose Knightsbridge it's London. You can open Euro account, or Sterling, or whatever. Service is great in my experience, and you can open it without visiting them (though good bit of paperwork needed). Internet banking of course, and they'll sort out the tax exemption.
HTH