That depends a lot on what type of bulb we are talking about (LED, incandescent, halogen, fluorescent, etc). For simplicity we (erroneously) assume that the bulb behaves like a linear resistor.
But because one of the bulbs has fused, would the new resistance depend on the other bulbs?
The new load resistance of all bulbs combined would be higher.
Would the current change accordingly (if so)?
The current delivered from the source would be lower.
The current through each remaining bulb will be slightly higher. This is caused by the internal impedance of the power source. Due to the lower source current, the voltage drop over the internal source impedance will also slightly lower, so the bulbs will see a slightly higher voltage. If the source impedance is very small as compared to the bulb resistance and/or the number of bulbs is large, this effect is negligible.