Bilingualism normally isn't a problem.
Dad was from Eastern Europe and insisted on speaking only English to me, for fear that I wouldn't be fluent in English. This was well-meaning, but misguided.
I later studied another Eastern European language at university level. I still grieve the missed opportunity to be bilingual. I've since picked up some of Dad's language and have taken lessons, but I'll never be fluent now.
The only way that being immersed in a non-English speaking environment could harm someone's ability to speak English would be if there was no equal exposure to English through school and the wider community.
Part of my responsibility in my old teaching job was ESOL. The only ESOL children who had difficulty picking up English were those who were not attending school regularly.
Our Eastern European pupils in particular used to get top marks in English and other subjects. In a previous school, a young woman from a South Asian background gained a place at a prestigious university in spite of the obstacles placed in her way by her parents. (She was "too western" so they packed her off to a boarding school in their home country. She quickly returned to Scotland: she wasn't daft - she made sure that she was expelled from boarding school within the first week.)
What I did see a lot of would be mothers from certain backgrounds who had learned no English whatsoever - in spite of living in the UK for years - and who had to bring their children to parents' evening to interpret. (Strictly speaking, they were entitled to an interpreter to be paid for by the council, but this seldom happened apart from admissions meetings and meetings concerning behaviour.)
However, I'm aware that certain communities used to try to game the system by insisting that their children born in this country were entitled to extra time in exams because they supposedly had English as a Second Language.
I don't know what rules apply in England, Northern Ireland and Wales, but when I was still responsible for ESOL in a Scottish school, you got an extra 15 minutes per hour (I recall) in order to give you time to access a dictionary in your "native" language. No dictionaries were allowed for English or MFL exams, so no extra time was available there.
I recall one pupil in particular who - every single ruddy exam diet from S4/Y11 to S6 - used to complain to the Chief Invigilator that they were entitled to extra time.
I'd be teaching a class (having already provided any dictionaries as required at the beginning of the school day) and the CI would be at my door asking for a dictionary for X. Every time I'd have to say "No. There is no entitlement. Yes, X is fluent in the parents' language. However, the pupil was born in Scotland, is fluent in English and cannot actually read their parents' language. Therefore, a bilingual dictionary would be of no use to them and they have no entitlement to extra time."
I also had a case where staff kept insisting that a brother and sister had difficulty accessing the curriculum because they had ESOL. They wouldn't believe me when I said "No...They were born here...They simply have difficulties of a global nature..."
In the end, I asked a TA who spoke their other language to hold a conversation with them. Her verdict: "They're just badly brought up. They haven't been taught to use respectful forms of the language when speaking to an adult. It's bad parenting."