This is interesting. According to
this map of Euston station the Northern line branches are respectively 20.4m and 29.8m underground, so by themselves that doesn't seem like an obstacle. I'm guessing that would comfortably leave room for an HS2 concourse - say, 5m below the existing National Rail platforms, and HS2 tracks 5m below the concourse - so going above the Northern Line branches and close enough to ground level for escalators etc. not to be too much of a problem.
I'm not sure the British Library and Francis Crick basements are such a problem in themselves: I would have thought the easiest solution would be to align the Euston HS2 platforms SW-NE, so that the Eastbound tracks pass to the North of Francis Crick. That also has the advantage of taking the lines comfortably out of the way of Kings Cross tube station (And even if you couldn't avoid them, would the cost of claiming either institution's basement and then paying to build another one somewhere else really be significant in comparison to the total cost of HS2? After all various utilities etc. had to be moved in order to build Crossrail, and that didn't stop Crossrail getting built).
However, you then hit a problem of threading across first the Thameslink tracks, and then the Piccadilly line (
21.3m below ground) and the Victoria line (15.7m - apparently the Victoria line climbs considerably between Euston and Kings Cross?). I don't know enough about the precise alignments to know how possible that is - I'm guessing it could be doable with some gradients on the HS2 lines to drop below Thameslink and then rise again to pass over the Victoria line. It would certainly be easier the further North you are because of the increased horizontal distance between the three northbound lines - that ties in nicely with trying to route North of Francis Crick.
Obviously, I realise the decision has been made so this is all hypothetical, and there are all sorts of questions about whether extending HS2 East would have been worthwhile. I'm more interested here in whether it would have been practically possible.
In terms of a use for it, personally I think it would have been nice to extend HS2 to Stratford, as that would make access to HS2 much easier for a fair swathe of East London and Essex, and would also relieve some pressure on both Crossrail through Central London and the tube network around Euston. It would also allow a connection to HS2 if at some point in the future that's considered desirable (though I think at the present time that was ruled out as not sufficiently beneficial).