I live in Co Durham and go to the Lakes all year round and winter conditions requiring axe and crampons are quite intermittent, so you could check the Helvellyn felltop conditions and the information given includes snow levels, depths, quality etc and whether or not winter mountaineering kit is needed (and the ability to use it!) If they are, you could go somewhere else.
You could, as you say,. go to Northumberland. If you go to the Cheviots there's only a couple of places where an axe and spikes might be needed and the hills are quite rounded. They are more remote than the Lakes Fells, though and, generally speaking you have to walk further and longer with more of a risk of running out of daylight than sliding off a hill. The Dales, Howgills and North Pennines can also provide good, rufty-tufty walking too.. and they're about the same distance from Co Durham as the North/East Lakes.
You might consider getting some khatoola spikes which will fit on to a wider range of boots than crampons and will be quite a bit safer on icy paths (not much good on steep ice, though) - you're just as likely to slip and hurt yourself on a small hill such as Simonside at Rothbury as a on a path on a bigger one, so having something spiky on the boots when its a bit skitey seems sensible to me...