In the pic below the West End/ City of Westminster area is in bright red, whilst the main financial district (the City of London) is in darker red next to it. Together they form much of the city centre, though strictly speaking its grown well beyond those medieval entities, having reached as far west as Kensington and Chelsea by the Victorian times and recently crossed the river and spread into the East End, following decades of cultural, business and construction booms (orange).
Although the cluster in the foreground (Canary Wharf et al) looks large it's not part of the centre - bear in mind the foreshortening affect of perspective, it'd appear much smaller if in the orange zone at the back:
To recap Westminster is a largely protected area of thousands of heritage buildings so not many highrises are going to impinge on it. Many of the tall building projects are pushed into the areas adjunct to Westminster, away from its protected sightlines, in outposts as far off as the Docklands, Vauxhall, Elephant & Castle, Stratford etc, these nodes of highrises fleshed out or forming that is making the city a multi-nodal entity, though they aren't strictly part of the city centre. There is currently a big impetus to change the city from facing west to facing east, where much of the transport infrastructure is becoming centred on.
In other words, the skyscrapers don't define the centre (protected Westminster et al), a lot of the development is outside of it on the 'approaches' and in satellite nodes. But they are in effect 'growing' the centre, knitting up areas into one day an ever larger whole, as befitting the large population growth (+120K a year). This is why the construction is booming, population pressure + property bubble + overdue housing crisis + office demand long constrained by heritage laws.