I am not a native English speaker either, but my ds's reading is excellent.
I haven't really read him bed time story either. (Not common thing in my culture, I don't have any experience of my parents reading books to me.)
I love reading, and read all the time, have books in my bag and read if I have a minute. He naturally copied what I do.
So, I don't think parents being foreigner prevents children to learn to love reading.
If reading English book is hard for you, reading children's books with your dd is great. They are not hard to understand, and can easily have conversation about it.
Even you work full time and very busy, I'm sure you can find 5 minutes a day to enjoy books together.
It's great if you can find somebody to help her once a week, but I think it's way better to do 5 minutes a day, every day.
There are many ways to get lots of books without paying fortune in England, even if you can't go to library regularly.
We got many books from charity shops, book people website, car boot sales, etc.