It seems that Warnock-Smith is basing his remark on the official Airports Commission forecasts (made in 2014).
I did find a forecast by OAG with this conclusion about the 3rd Heathrow runway:
In our cautious assessment of the opportunities, we project that over half of
the new capacity could be used within two years from opening, if not faster
allowing for existing carrier expansion. For such a large capacity increase
to be absorbed so quickly it highlights the case for new runway capacity in
London and the South East. It also suggests that we should already have a
third runway and perhaps be planning a second elsewhere!
It does have some 20 pages of forecasting by segment (destination) before coming to this conclusion.
I see a weakness that it's based on the current EU Airport Slot Regulations at least for connections with the EU, so who knows what will happen with Brexit etc. They don't analyze that scenario.
Also, this OAG model is unconstrained, in the sense that it assume capacity at corresponding airports elsewhere will increase keeping up with demand.
The Airports Commission has a much more detailed (and technical) forecast using various scenarios, including constrained capacity elsewhere, and considering other factors like carbon-caps. It's quite a long document and a lot of it is focused on mmpa (million passengers per annum) forecasts rather than runway capacities, but it does say this on p. 125:
By 2050 the additional 260,000 ATM [Air Transport Movement] slots from the new Heathrow
capacity has been fully utilised for many years in all the demand scenarios. [footnote: Ranging from 2032 in global growth to 2042 in global fragmentation.]
These are two scenarios they ran. So it does seem that the answer (of Heathrow filling up again in 25 years) is yes in these Airports Commission forecasts. The 25 years number seems to coincide with the 2042 (-25 = 2017) forecast, so it's perhaps a rounding of a real forecast, not pulled from thin air.
Also earlier in this long paper (p. 104):
The timelines illustrate that, if carbon is capped, the new runway at Heathrow is fully
utilised by 2034 in low-cost is king and by 2040 under the other four demand scenarios.
By 2050 London City with its short-haul network would be fully utilised under three of the
scenarios but none of the other 3 London airports would reach full capacity by 2050 under
any of the five scenarios.
These scenarios have assumptions that are detailed over many pages (pp. 38-48), so I'm not going to reproduce them here.
Heathrow Airport Limited (HAL; the promoter of the Heathrow Airport
North West Runway option) has actually written a critique of these government forecasts, claiming that they are too conservative; HAL claims the demand growth would be even faster.
Both of these latter documents (Airports Commission, HAL) were written/published around 2013-2014, so there is the issue of them not anticipating Brexit.