Not if you manage to ride the bike up at a constant speed. Compared to riding on a windless day on the flat, the principal difference is that you will be lifting yourself + bike uphill, gaining potential energy (the formula is dead simple - i.e. mass x g x elevation). Of course in reality it is not possible to ride at a constant speed up the 21 bends hence the 1.8L in the tyre should make a difference - It is however imponderable exactly how much difference could theoretically be attributed to the water in the tyres without a lot more data.
However, it is quite easy to evaluate theoretically and fairly accurately how much slower normal additional mass can cause for that or any climb. Let us use his normal bike scenario, assuming no wind, and given he was climbing Alpe d'Huez which is roughly 13km with a total elevation of 1km. Since he made it in 49' 40" in standard trim, and since it takes about 30W to travel 13km on the flat in a windless day over that time, we can evaluate the mass of him plus his bike, because using the potential energy formula above (and energy is power x time), power consumed to combat the vertical elevation alone is 278W-30W or 248W = mass x gravitational constant at 9.8ms-2 x elevation 1000m / time of 2980 seconds it took him. Hence the mass of him and his bike is give or take 75kg, not an unreasonable number huh?
If we add another 1.8kg to his mass like his water bottle, using the same formula and power, it is also easy to find that the new time = mass of 76.8kg x gravitational acceleration at 9.8ms-2 x elevation of 1000m / 248W, or 3035 seconds which is 50 minutes 34 seconds, i.e. 54 seconds slower. A bit less than his measured difference of 1 minute 54, but given power meter accuracy is not great, there might be wind etc., not too bad.
What the above really shows, is it is really not difficult to evaluate the time impact of just simple mass on climbs quite accurately.