Which were the last castles to be besieged and fall in the United Kingdom? I'm separating the two in case a castle was besieged unsuccessfully most recently.
Fort Augustus was besieged on 3 March, 1746, and surrendered two days later. If not the absolute last, this was certainly one of the latest successful sieges in Britain. This followed an earlier action in December 1745 when Fort Augustus was captured by government-aligned militias.
If we were to be picky about the name, the last successful siege of a placed styled 'castle' occurred slightly earlier. Inverness Castle was besieged on 17 February, 1746. The garrison also surrendered within two days, and the castle was then demolished by the victorious Jacobites.
Another notable example was Carlisle Castle. Government forces laid siege to it on 21 December, 1745, and the Jacobite garrison surrendered nine days later. It had previously been taken by the Jacobites on 15 November, after a two day siege.
These all occurred during the last major Jacobite uprising, which broke out in 1745 and continued into spring the next year. On 13 July 1745 the Young Pretender landed in Scotland, seeking to regain the thrones of Great Britain that were his forefathers'. He successfully raised an army to defeat the only government force in Scotland on 23 September, and proceeded to march on Carlisle - one of the most heavily fought over places in Britain, owing to its proximity to the Border.
His invasion of England soon faltered, however. With military support from France not forthcoming and a largely unenthusiastic English populace, the Jacobite army was forced to retreat back to Scotland in December. They won a minor victory on 17 January 1746, but in April the Jacobite cause was decisively defeated at the Battle of Culloden.
By 1922, the Irish War of Independence had ended with the Anglo-Irish Treaty. Most of Ireland was then placed under the authority of the Provisional Government, although technically the Irish Free State wasn't established until December that the year, at which point the United Kingdom adopted its present name. I'm not sure whether this transitional period counts as part of the UK.
Anyway, it should be noted that the Republican seizure of Killkenny Castle on 2 May probably doesn't count as a successful siege. They apparently walked in and took the castle by storm in the early morning, i.e. without laying siege. However, the occupiers were then besieged by forces loyal to the Provisional Government, and surrendered the next day. So the last successful siege is on 3 May 1922, when the Free Staters retook the castle after a 1 day siege.
In May 1922 Republican forces took over Kilkenny Castle.
The whole Island of Ireland was part of the UK until December 1922