Be consistently one parent one language. We live outside the UK. I speak English to the kids, DH speaks Dutch to them, DH and I generally speak English to each other. The kids choose which language to speak to each other depending on their mood.
My eldest can't write very well in English but taught themselves to read and now devours English graphic novels. My youngest is at a bilingual school (which is a benefit to the other kids English having a native speaker in the class and helps boost his confidence) and can write and read in English as well as in Dutch.
When DC1 was little I bought the Oxford reading books to support his 2nd language learning at home (though I waited until he could read in Dutch first).
I've always been consistent in making sure they speak to me in English since for most of the time I've been the only one speaking English with them in their daily lives. If they would start to talk to me in Dutch I always remind them to speak in English, or I repeat back what they've said in English to them. When they started school they obviously were encountering more vocabulary that they only had in one language and this was where we'd talk about what they'd done at school, encouraging them to ask, "how to you say X in english"?
My kids do hear me speak Dutch now, especially when they have friends round who don't speak much English, but I was super strict with being consistent when they were younger. This is hardest when they're tired. They do like to correct my Dutch as well.
My kids read and watch films in English by preference (I have to force DC2 to read in Dutch at times!). They tend to watch TV in Dutch. I also make sure they have access to both Dutch and English magazine subscriptions, and Dutch/English books (library, or bought, often 2nd hand)
I also have some friends who've had the same experience you mention. There was no other way for them to connect with their English speaking relatives and eventually they'd come round as long as their English parent kept talking. In other cases it can be that the child feels their other language is not useful or respected - it's up to the parents to keep it alive, learn with them, stimulate their interest.
My Dutch MIL asked me why on earth I'd want the kids to be able to speak English when they live in the Netherlands - now she's always boasting about her bilingual grandkids.
I read the following book when I was pregnant with DC1 which I found helpful with lots of practical activities/tips: Growing up with two languages: A practical guide for the bilingual family by Una Cunningham.
https://www.amazon.co.uk/gp/product/0415598524/ref=ppx_yo_dt_b_search_asin_image?ie=UTF8&psc=1
Good luck! It's been hard work to get where we are (DC are 12 and 9) but as someone who has been monolingual for most of their life, I really think it's an amazing gift you can give your children. DC2 and I are learning another language now because their uncle is from the country and their cousin will also be brought up bilingually in his language + English from my sister.