Family are "technically" Jewish

I’m from the UK, and after visiting the Jewish Quarter in Prague, some questions about my family history came up. My mum’s family migrated from northern Germany to the UK in the early 1900s, partly to help build infrastructure and partly to escape the growing antisemitic rhetoric in mainland Europe. A family friend, who is a university lecturer familiar with this history, confirmed the story when my mum mentioned it.

My mum’s last name is very rare in Scotland—her family might be the only ones with it—and it was changed during World War I because it sounded too German. We’ve always known about this aspect of our history but never really looked into it further.

Given this background, how Jewish does that make my family? I do get that it’s up to interpretation, but I’d like to hear what people think. And if I wanted to connect more with the religion, how would I go about it? I realize it’s been 100~ years since the migration and don’t want to sound like an American claiming they’re “Irish”, but I’m genuinely interested.

tldr: family are technically jewish, how jewish does this make me