E. Nesbit & Edward Eager

When my daughter was six, we read all the Edward Eager books together. As she got older, we read a few E. Nesbit titles. Other British/Irish authors we've read include Susan Cooper, C.S. Lewis and J.R.R. Tolkein.

Can anyone recommend something strongly that might fit more or less into this vein?

Try Alan Garner. Weirdstone of Brisingamen and Moon of Gomrath should fit the bill.