Reunification meant that Germany's power in the EU increased. Becoming a bigger country with a larger population got more seats and pushed the bloc more in its direction. Thus reducing Britain's influence there.
It has to be said that the matter was somewhat settled in European circles by an agreement between Francois Mitterrand and Helmut Kohl: Germany would be reunited, but in exchange it would be "bridled" by the common European currency that was being designed. Most other countries were in support of this arrangement, since they stood to gain from effectively sharing the power of the almighty Deutschmark (which they did, and are, despite some other issues). Britain had no realistic chance to oppose anything.
And ironically had Britain joined a currency pulled weaker but the eurozone's expansion, they might have been able to stay an industrial power like Germany!
Was German reunification setting a president for Irish reunification never a concern for the UK government, like how e.g. today Scottish independence is today a concern for the Spanish government setting a president for an independent Catalonia?
Perhaps, but I don't think it was much of a realistic concern for any who held it in the UK government.
Whereas Germans, East and West alike, had largely thought of themselves as Germans in varying senses since Bismark (and both the DDR and BRD claimed to be legitimate successors to the same historical state), the peoples who lived (and live) in Northern Ireland didn't all think of themselves in the same way. Some thought of themselves as British, some as Irish, and many as some mix of both. A big reason why historians mark the Good Friday Agreement as the end of The Troubles is that it recognized the right of each person in Northern Ireland to identify with the nationality they chose (self-determination), in a way that both the British and Irish governments cooperated to recognize in administrative affairs where nationality mattered.
There wasn't a separate ethnoreligious East German identity. There were (and are) conflicting identities in Northern Ireland with ethnic, religious, national, and historical dimensions. The two situations on-the-ground were extremely different.
Why would the UK have even pretended to want to stop it? Weren't their foreign policies, at least towards communism, heavily aligned with the US?