The immigration problem would be likely to arise for a person who needed a visa to enter the UK.
The UK Immigration Rules (Part 1, no. 15)
state:
15. The United Kingdom, the Channel Islands, the Isle of Man and the Republic of Ireland collectively form a common travel area. A person who has been examined for the purpose of immigration control at the point at which he entered the area does not normally require leave to enter any other part of it. However certain persons subject to the Immigration (Control of Entry through the Republic of Ireland) Order 1972 (as amended) who enter the United Kingdom through the Republic of Ireland do require leave to enter. This includes:
(i) those who merely passed through the Republic of Ireland;
(ii) persons requiring visas;
(iii) persons who entered the Republic of Ireland unlawfully;
(iv) persons who are subject to directions given by the Secretary of State for their exclusion from the United Kingdom on the ground that their exclusion is conducive to the public good;
(v) persons who entered the Republic from the United Kingdom and Islands after entering there unlawfully or overstaying their leave.
By "merely passed through the Republic of Ireland" is meant, I believe, the situation in which the traveller did not clear Irish immigration. (However, although this provision exists, the situation does not arise as airports are presently set up, since, as noted above, there is no routine immigration control between the Republic of Ireland and Northern Ireland and anyone transferring from an international flight on to a Belfast-bound flight would pass through immigration.)
There are, however, sometimes ad-hoc controls at the border, so if you do require leave to enter the UK (i.e. you are not a British citizen, a Commonwealth citizen with the right of abode, a citizen of an EU/EEA country or Switzerland, or a person with leave to remain in the UK), then it would be sensible to have details confirming your length of stay in the UK, etc, to hand. If you fly between the UK and Ireland, you are more likely to have to show your passport to an immigration officer travelling
to Ireland than
from Ireland (certainly for trips between Great Britain and the Republic).
If you require a visa to enter the UK (i.e. if you are a visa national or if you are coming for other than a short-term visit), you will need to have a UK visa (an Irish visa would not suffice) and it may need to be "activated" after arrival in the UK.
Assuming that you are not a visa national and don't need a visa to enter the UK for some other reason, then you ought to be fine, but the Irish authorities will (I
think) give you permission to remain in Ireland (and therefore in the Common Travel Area) for 3 months, rather than the 6 months' leave to remain that would normally be granted at a UK port of entry. Obviously, if your stay is for less than 3 months that is not a problem.
Changes are being proposed to provisions for travel between the UK and Ireland: whether any changes would make it all more straightforward remains to be seen!