I upvoted Normille’s comment to offset the downvotes. It may have been crudely put but it’s not wrong – at least in the case of Ireland, the country I know best.
For as long as I can remember, the Irish government has been quite deferential towards the US, e.g., allowing the use of Irish airports (mostly Shannon, a civilian airport – but also Casement Aeorodome, the headquarter of the Irish Aer Corps) by US military to carry troops, weapons and – quite possibly (aircraft were never inspected) – victims of “extraordinary rendition”. Every year, our Taoisigh (prime ministers) are always eager to have a photo opportunity with the US president on St. Patrick’s day.
I’m guessing that this is mostly to attract Foreign direct investment into Ireland from US multi-national corporations as can be seen by our notably low rate of corporation tax – and the government’s unwillingness to accept the tax income that the EU considers to be owed to us by Apple. Neo-liberalism has been the dominant political ideology over the past few decades with one former Tánaiste (deputy prime minister) famously declaring that Ireland was (ideologically) closer to Boston than Berlin. This dominance left Ireland particularly vulnerable when the 2008 financial crisis eventually hit us – without the widely touted “soft landing”.
However, the relationship isn’t all bad: a number of US administrations – particularly Bill Clinton’s – helped to bring about the end of armed conflict in Northern Ireland. That was probably the most important political achievement in the recent history of Britain and Ireland (one that Brexiteers, sadly, don’t seem to care about or have forgotten about).