The result of mass migrations of foreigners is welsh culture too. Just a different mix, and at different times from the rest of Britain. Scotland, Cornwall, Greater London, The North East, The West Country, and East Anglia also have a noticeably different mixture to the rest of Great Britain; Indeed, each town and city has a slightly different culture - and in some cases, fairly small areas have huge differences. North Wales and South Wales have little in common, and it is only in recent years that people in South Wales have started speaking Welsh again, after the Welsh language almost completely died out in that part of the country. In Newcastle, the dialect (which they claim is of English) spoken is incomprehensible to people from the South, and owes more to Old Norse than to Old English, or Norman French. They even use words and phrases that survive today in Denmark and Norway, but not in London. I have met people who live only a few dozen miles from my birthplace whose speech is almost impossible to comprehend.
The Welsh and the Scots have kept a legal border separating them from England; There are no such borders separating Newcastle, Liverpool, Birmingham and London, but people from any of those cities have trouble deciphering the speech of people from the other three.
British culture - indeed, just the English part of British culture - is already a patchwork of very different peoples, with differences in language and attitude. Adding a few more disparate groups to the mix is not a particularly big change; nor is it unprecedented.
