This is a good question, the answer to which would surprise many people living around the world.
Ignoring the many languages spoken in the UK by immigrants, there are several native languages of the British Isles, which are as follows:
Cornish
Spoken by around 2000 people in Cornwall, south-west England
Gaelic (Irish)
Spoken by 1.7 million in Ireland and around 100,000 in the UK (mostly Northern Ireland)
Gaelic (Scottish)
Spoken by around 60,000, mainly in the Outer Hebrides
Manx
Native language of the Isle of Man, it went extinct in the 1970s but has been revived and is now spoken by a few hundred people
Welsh
Spoken by 750,000 in Wales and England
Norman/French (Guernésiais, Jèrriais and Sarkese)
Spoken by a few thousand on the Channel Islands
Many other native languages have since have gone extinct over the centuries.
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Other languages spoken in the British Isles, which are not strictly-speaking native languages, include:
English
The most commonly spoken language of the British Isles. This is a mongrel language, a mix of Germanic (Saxon), Norman/French, Nordic and Old English, all brought to the Isles by various waves of invaders. Over time Modern English has superceded the native languages in most of the British Isles.
Scots
Germanic language, a variant of Old English, spoken in the Scottish lowlands; number of speakers unknown