Use NCN862 from the A580 (Knowsley) to Croxteth Hall & Country Park.
Ah, I wasn't aware of NCN862, though I do unknowingly use part of it when I tootle round Croxteth Hall Park (I live fairly close), and that's quite pleasant. I use the Gellings Bridleway and Littlewood Greenway parts. If you go that way, I'd suggest modifying the approach into Croxteth Hall Park a little myself, as the more direct route was very very muddy the last time I tried it. Here's a Strava track of my last ride there...
I was on my MTB at the time, so I've marked a suggested road bike route with blue arrows.
Also, if you wanted to get straight into the city centre after coming out of Croxteth Hall Park instead of heading south on the Loop Line (though it's worth popping in to the Loop Line at that point to see West Derby station if you like railway things), the direct route from there is pretty reasonable - I use it to get to town myself. It would save doing that reverse section along the river front.
Up Mill Lane (the only tight bit, narrow lanes and often heavy traffic, but it's very short), across Queens Drive and down Mill Bank, along West Derby Road (A5049) and then duck into Newsham Park just after the railway bridge. Diagonally across the park to emerge on Sheil Road (B5188) then follow Boaler Street and Farnworth Street, cut the corner across Phythian Park, on to Kensington (use the bus lane), Prescot Street, and you're pretty much in the city centre.
Give it a trial run on Google Streetview and see what you think.