British, Commonwealth, or Irish passport holders may enlist with the British Army (must have been resident for at least 5 years prior if you're planning on becoming an Officer).
To apply for naturalisation you need to be officially resident with a visa for Leave to Remain / Right of Abode / Settlement to join spouse/parent/children for at least 5 years prior (3 if you're married to British citizen).
As an EU citizen you obviously don't need visas or leave to remain but you DO need to be able to prove that you have been resident in the UK, with your main estate and taxes being paid in the UK for at least 5 years (a few hundred days spent out of the UK in that period is generally accepted). The 12 months before your application for naturalisation must have been spent resident in the UK.
On top of that you will also need to somehow demonstrate 'significant ties' with the UK. This is really up to the whim/discretion of the immigration official reading your forms but typically accepted ones are marriage (obviously), having British children (obviously), ancestry, or long / conspicuous service to the Crown.
edited for the bleeding obvious
