Information from a council planner about the height of the proposed building on the Brill Place site (the land behind the British Library):
The indicative material in this respect would have shown a building consisting of up to 5 storeys with further levels of roof plant above. … the laboratories will entail far greater floor-ceiling heights than normal residential and the equivalent residential height will therefore likely be more in the range of 8-13 storeys.
The development control forum, which will be the last chance for residents to comment on the plans before a development application is submitted, is likely to be on February 10. More firm information here when available.
More about the UKCMRI.