Yes and no. Yes, there's a hell of a lot of "green", in fact isn't something like 90% of the UK not built upon, but a lot of that is land you can't build on, such as marshland, moors, bogs, mountain sides, steep valleys, etc. And in most of it, there isn't the basic infrastructure to support major development, i.e. water supply, sewerage, gas, electric, telecoms, roads, public transport, schools, GP surgeries, shops, etc. It'd be a major task to build, say, a new town with thousands of homes in the Yorkshire Moors, or the Lake District, probably akin to the costs and timescales of HS2! Lots, if not, most of the houses in the more rural/remote areas are powered by calor gas (no mains gas), and have septic tanks (no sewerage systems), some even get their electric from generators. What infrastructure there is in sparsely populated rural areas simply isn't adequate to cope with hundreds of new houses.
In reality, new homes are built on "greenfield" just outside existing major towns and cities so it can "tap into" the existing infrastructure.
And the UK isn't short of just "a couple of thousand" new houses - estimates vary, but it's often said to be hundreds of thousands if not 2-3 million new homes we need! We don't even have the workforce to build what we need!
We need a national plan to increase the number of homes for people to live in. Only PART of that is building new homes, other aspects need to be disincentives for holiday lets, repurposing/rebuilding of city centre retail/industrial areas into housing, incentives for older people to downsize from big houses, forcing local councils to compulsory purchase empty homes, encouraging students to live at home and go to local universities. There is no single answer to the problem.