To start with, you're talking about
electricity generation, where he is talking about total energy requirements. So your 15% figure doesn't include the energy consumption of cars, aeroplanes, many types of heating etc.
I believe he also includes the embodied energy of imports (this was in his book), where if goods are manufactured overseas and imported, it may make sense to count the energy used to manufacture them, depending on what you're measuring exactly.
For example, there's little difference between importing aluminium, or importing energy and bauxite and smelting it yourself.
Also, area use will not increase linearly with the addition of many renewables, because in many cases the best sites are taken first. Hydro can generate a lot of energy around the clock, but you quickly run out of the best locations. Wind strength varies across the country.
In a small country like the UK, solar isn't going to vary much by latitude, and you can probably build it out all over the country. But time shifting of demand isn't always possible, so even if the panels were free, using them to push from (say) 50% of energy from renewables to 100% is going to involve some kind of storage system, which is either going to be very expensive (probably exceeding GDP) and unproven (eg hydrogen fuel cells, lithium ion batteries), or require a ton of space and specific terrain, like pumped water. So that's going to cost a lot of space.
Lastly, electricity isn't always the desired final form. If you're powering jet planes, you're going to need some type of biofuel, which is very inefficient per unit area. If and when hydrogen planes are developed, this will stop being an issue (electric cars have already proven their viability).
I highly recommend his book. It's a brilliant read, he nicely boils down the issues, and is thoroughly committed to stopping global warming - it's not some doom and gloom oil industry sponsored thing. But he doesn't have his head in the sand about how hard a switch to renewables is going to be, unlike (unfortunately) a ton of the green movement seems to.