As usual in these matters Lord B is sadly mistaken.
It would be possible, just, to get the high speed line up at Queens Park through the space released by the builders yard. However there would definitely be land take; as ever you can’t just magic major infrastructure into existence, someone has to build it. And to enable building you need land, and lots of it. Have a look at the land take at other tunnelling sites on HS2.
It would also be rather disruptive to the most valuable section of the WCML.
And, also, you cant get all the freight via Hampstead Heath, and the DC line tunnels would need reboring to take OLE.
(I know this because I’ve looked at it myself personally in the course of pre HS2 studies).
Land take/ civils working area is something which always annoys me in my current (non rail) project. From a first principles perspective it should be possible to build things very close to each other, in practice heuristically the civils people will point to examples of insufficient working space leading to delays.
The key point is that civil construction projects are orders of magnitude away in terms of cost and efficiency from where they could be theoretically if they were more like volume production industries. Ergo if you planned to activity down to the level of modelling the individual workers, used large amounts of automation, tooling and off site manufacture you probably could build stuff like the concept originally described in a tight area and with minimised disruptions to ongoing activities. (I am aware that all those things are current buzzwords in construction however the industry is very conservative and is nowhere near fully exploiting them)
However nothing like that will happen if you are set up as a project like HS2 is, if "Britain's New Trunk Railway" was set up with the goal of connection GB's top 20 conurbations to highspeed rail and then building each of those conurbations a mass transit/regional rail system, plus transit orientated development with a set budget of ~£10 billion a year (potentially much less if they are allowed to raise revenue from property) and finish point of ~2060 then you would be in a position where you could invest in owning your own capabilities to do things, have your own R&D department and learn by doing over extended time periods.
While we are at it BNTR should have pretty broad powers to set it's own route on the basis of "net public/environmental good" with a broad requirement to fully investigate alternative routes. We need to stop trying to engineer around public perception problems because that then causes the biggest public perception problem which is massive costs. HS2 has a vague goal, build a fast railway to Birmingham and maybe Manchester, sometimes it's for capacity sometimes its for speed and a massive vague price tag for something which doesn't seem that big and which most people can't imagine using. If the goal is much more ambitious like build a high speed railway to every big city, plus mass transit more people are likely to be on board even if it won't get to them for decades and 0.5-1% of government annual spend to do it is again difficult to make the "HS2 will bankrupt the country" statements against.
Really we should have started "BNTR" off by doing something like a Leeds-Teesside-Newcastle High Speed Railway learning all the lessons on a relatively easy bit, under current governance you couldn't do that because it would be likely that such a project would have a negative cost benefit on its own. We need to get the treasury out and embrace ambiguity over the long term, sell the project as a fixed annual budget and variable end date with the proviso that as we get better at doing this over time we will get more done per year as the project continues.