For the hypothetical case of a thermally perfectly insulated system, I'm sure you can work out yourself from the specific heat and the enthalpy of fusion for water. Given that the enthalpy of fusion (330 kJ/kg) and the specific heat of ice (2 kJ/kg-K) have a ratio of 165 K, and you need the entire ice bucket to stay below the melting point, and your 20:1 ratio of ice/water, I can give a quick estimate that it will definitely not freeze down for an ice temperature above -165/20=-8.2 K. (I neglect the heat to cool down the water from 21 °C to 0 °C).
For the practical case of heat entering from the outside world into the bucket, it gets much more complicated. What you need is the heat-transfer coefficient from the ice cubes into the bottle and the heat-transfer coefficient from the environment into the bucket. Once you know those, you can write a system of coupled differential equations (temperature of the ice cubes, temperature of the bottle, fraction frozen in the bottle) and you can find out how quickly the ice cubes reach the melting point versus how quickly the bottle cools down. You have not provided enough data to estimate either of the two heat transfer coefficients. In particular, the heat inflow from the environment will depend on the presence of air circulation and on the humidity - condensation of ambient humidity will result in much more heat transfer than in the case of dry air at the same temperature.
The heat transfer coefficient from the bucket to the bottle will be on the order of $h=\lambda/D$, where $\lambda$ is the thermal conductivity of ice and $D$ is the diameter of the bottle. (The effective thermal conductivity of ice cubes with air gaps is another difficult thing to estimate.)
Heat transfer from ambient to the bucket is much more difficult to estimate. For free convection (air circulation only due to temperature differences) combined with thermal radiation, $h=10 \mathrm{~Wm^{-2}K^{-1}}$ is a typical value. For heat transfer with forced convection and/or condensation, you'll have to consult a book on heat transfer. Even then, your question would probably require computer simulations for a reasonable answer. It's easier to set up an experiment. My gut feeling is that you won't get the bottle to freeze down even if you start with freezer-temperature ice (-18 °C, 0 °F) because I know from experience that freezing a 1L bottle of water in the freezer will take at least 12 hours and the temperature in your un-insulated bucket will rise to above -8 °C much faster than that.
Update
If you want to get some feeling for how to calculate it, you could proceed as follows.
Find an estimate for the heat transfer coefficient due to the combined effects of condensation, convection and thermal radiation. These three processes are additive; the convection and radiation will be around 10 W/(m2K). As for condensation, you might estimate it by putting a bottle of frozen water (-18 °C) out and measure how quickly it accumulates ice/water at the surface. Divide by area, multiply by enthalpy of vaporization.
Estimate the thermal conductivity $\lambda$ of the ice-cube/air mixture. This will depend on how they are stacked. Let's say, half of the conductivity of solid ice.
Calculate the thermal diffusivity, $\alpha=\lambda/(\rho C)\approx$6e-07 m^2/s. Estimate how long it will take for heat entering from the outside to travel to the middle (where your bottle is), using $R=\sqrt{2\alpha t}$, where $R$ is the radius of your bucket.
Estimate how long it takes for the water to freeze as a function of temperature at the bottle wall (assume that the ice is cold enough not to melt). For this, assume an effective heat transfer coefficient $h=Nu\lambda/D$, where $Nu=4$ is a typical Nusselt number for a cylindrical bottle of diameter $D$. ($D$ and $\lambda$ here are different from those above for the entire bucket)
Now you have to combine the data from the steps above. For a given initial temperature, is the bucket time scale (step 3) fast compared to the bottle time scale (step 4)? Then assume that the ice bucket changes temperature as a whole. From there on, it is relatively straightforward Is it slow? Then you can neglect the heat transfer at the outside of the bucket; you have to solve the heat transfer equation only near the bottle. Even if you assume cylindrical symmetry (temperature only dependent on radial coordinate), this will probably require a numerical analysis program to solve.
