Mainstream Lenders and time at address

I've recently moved to a new house having moved from the Republic of Ireland to Northern Ireland.

The only lenders I really stood a chance with were those with cross-border operations such as Danske Bank, Bank of Ireland and Ulster Bank.

Now that I've got a mortgage, how long do I need to wait before the mainstream lenders consider a remortgage?

I've read 3 years from arrival in the UK for people getting a first mortgage. My current deal is a 2-year discounted deal. Do you fancy my chances with the likes of HSBC or the Co-Op after 2 years assuming a squeaky clean credit history?

I realise that things will be different when the time comes - I'm just interested in the current position.