A capacitor is said uncharged when its voltage on its terminals is zero.
When the transformer primary is connected to mains, the equivalent impedance (internal resistance and leakage inductance of the transformer that are seen from the secondary terminals) limits the current that charges the capacitor. The initial charging current is relatively high, and as you said, it also depends on the initial instantaneous voltage applied on the transformer at t=0.
Also the dynamic resistance of the diode bridge being in series with the secondary coil contributes in limiting the charging current. But, practically speaking, the diode resistance is usually negligible in comparison to the internal impedance of the transformer.