Plants flowering on New Year's Day, apart from winter-flowering species, are perennials and late surviving autumn-flowering annuals. All else being equal (which it probably isn't) the commoner a species is the more likely you are to find it flowering on New Year's Day. Hence hogweed and cow parsley and (lawn) daisy and dandelion are perennials you have a good chance of finding in flower on New Year's Day. Typical annuals are mayweeds and sow thistles.
For examples of all things not being equal I don't expect to find cleavers (an early summer-flowering annual), or nettles or brambles or plantains in flower on New Year's Day. (Nettle and great plantain are not evergreen, which might explain their absence - cow parsley and hogweed aren't evergreen either, but cow parsley is wintergreen.)
Blueberries are rare enough that finding out of season flowers might be difficult, but rhododendrons regularly produce a few autumn flowers. You have me wondering about bilberries.