Your bipolar transistor is driving the lamps in a voltage follower fashion, meaning the output voltage can't be higher than the controlling voltage.
The basic rule to turn an NPN bipolar transistor ON (or just bias it) is to have about +0.6V on its base relative to its emitter, meaning the emitter can't go higher than the base potential minus 0.6V, in this case 5V-0.6V=4.4V.
simulate this circuit – Schematic created using CircuitLab
In order to apply the "full" supply voltage, you need to connect the bulb(s) on the opposite end of the transistor, that is, to the collector pin.
In that case, the emitter is connected to ground, and even a 1.0V logic or control voltage can turn the transistor on fully for whatever output/supply voltage (up to the transistor's withstand voltage limit, of course). This type of a circuit is also used in logic level translators due to this fact, as the output can swing from 0V to the full supply voltage.
The collector can now reach almost 0V (almost at the ground level), so if the bulb is connected to the positive end of the battery on one side and to the collector of the transistor on the other side, it will have almost the full 12V across it (let's ignore the voltage drop across the transistor for the sake of simplicity).
Here is the way it's done:
simulate this circuit
The R1 resistor serves to limit the current through the base which acts like a diode (base-emitter diode) and would burn without the resistor. The base-emitter junction stays at about 0.6V or slightly more while the transistor is on.
The collector-emitter junction now acts like a switch, open with no current through the base, and closed (ON) with a sufficient current running through the base.
Because a bipolar transistor has significant voltage drop across it while conducting a significant amount of current, it will dissipate a lot of wasted energy as heat and possibly burn out. At 0.5V across it and a 10A current, that is 5W of wasted energy and heat that needs to be removed from it.
You are also wasting energy to keep the transistor ON via the current running through the base. If you're running 10A through bulbs, you may need 1A through the base, which is not only additional 0.6-1W of heat added to the transistor, but a full 5W of energy needed from the 5V supply.
A much better and more modern solution would be a MOSFET transistor as a switch. It needs practically no current to keep it ON, while having a very small resistance and voltage drop across it.
Here is an example schematic:
simulate this circuit
As you can see, there is a very small voltage drop (a few times smaller than for a bipolar transistor) across a MOSFET when it's turned on. This means there is a few times lower loss and a few times less heat dissipated in a MOSFET.
Additionally, almost no current at the controlling leg (gate) means that basically no energy is needed to keep a MOSFET on.
For example, while you may need from at least about 50mA and up to about 2A of base current for 10A at the output with a bipolar transistor, you would only have a gate leakage current of 0.0002mA (0.2µA or 200nA) in a typical MOSFET at the most. That is less than the self-discharge current of a typical CR2032 3V coin cell!
To put it another way, a 3V coin cell could keep a MOSFET turned on for years, while with a bipolar transistor the same amount of energy would barely last a few hours at best (this could be improved with a "darlington" arrangement, but it would cause higher output losses, and it would still not come close to the MOSFET efficiency).
The included MOSFET model is just an example, you could use one with a lower resistance while on (called RDS(on)) to have an even lower voltage drop across it, or use a logic-level MOSFET which works better when you don't have 7-15V typically used to turn it on.
The only drawback to using a MOSFET is the voltage required to turn it ON needs to be significantly higher than for a bipolar transistor, at least 3-4V though typically 7-10V vs. 0.5-1.0V
You could also use an 8V to 10V regulator (LM7808 to LM7810) to supply the NE555 (which can run at up to 16V at most, but uses more quiescent current with higher supply voltage, unless it's a CMOS version like TLC555 or ICM7555), which would let you use any MOSFET that has the lowest RDS(on).
For the sake of simplicity, I have kept the answer to the most often used and practical examples, but you can also use a PNP bipolar transistor or P-channel MOSFET (the above was an N-channel).