You only pay CGT on a house if it's not your main residence - that is, if it's an investment rather than a home. There certainly are plenty of landlords in London, but i suspect that in the areas where Crossrail is increasing prices, it's mostly homeowners.

Won't they be paying Stamp Duty?

You pay stamp duty land tax (SDLT) on the purchase of land or property, not sale.

But for every sale there is going to be a purchaser so the tax is going to get paid whether it is the seller or purchaser paying.

Sure, but arguably thats penalising the wrong party. Also depends on who purchases, if you're a first time buyer you get significantly reduced stamp duty for example.