score:5
No problem, you don't need to get any stamps. As long as you can legally enter both the UK and Ireland, all is fine.
Ireland and the UK are part of the Common Travel Area (CTA), which means passengers moving between them are not always checked.
When entering the UK, you're entering the CTA and can travel freely within it. You may want to explicitly state that you intend to visit Ireland.
When leaving Ireland, you're expected to have proof of entry into the CTA. Your entry stamp into the UK will prove this.
Upvote:1
Actually non-EU/EFTA citizens are supposed to be stamped when entering Ireland from the UK, though not the reverse.
This is because the visa-free stay in Ireland is 3 months, and in the UK 6 months. The UK doesn't grant separate admission if entering from Ireland, however, so the 3 months granted if entering the CTA through Ireland includes stays in the UK as well.
If entering by air from the UK, rest assured you'll be checked and stamped. Same when entering by Irish Ferries in Dublin as a pedestrian, but if using a vehicle or the London-Dublin bus, the Stena ferry or a bus/train Northern Ireland-Ireland, only spot checks are performed.
In case you're not checked and stamped, you should go to the immigration office in Dublin to get an entry stamp, or, if entering from Northern Ireland and being able to stop by in Dundalk, to the police (garda) station in Dundalk.