The sawtooth is actually positive and negative exponential decays, not lines. However, for many purposes this is close enough to a sawtooth.
To visualize this better, let's do a example with hard numbers. Let's say this is a ideal Schmitt trigger inverter running from 5 V with thresholds at 2 and 3 volts. Let's start with the input at 0 (the cap discharged). This will make the output high, so now the cap charges up with a exponential decay towards 5 V. When it gets to 3 V, the inverter output will go low. Now the cap discharges with a exponential decay towards 0. When it gets to 2 V, the output will go high until the cap reaches 3 V.
This process of charging, discharging, charging, etc between 2 and 3 volts will repeat indefinitely. The charge and discharge times are the same, so the output will be a square wave.
Each charge and discharge is terminated 1/3 of the way to the final value, which is .41 time constants. With the values you show of 1 µF and 100 Ω, the time constant is 100 µs. Each phase will therefore last 41 µs. The period will be 81 µS, and the frequency will be 12 kHz.
Keep in mind that the 2 V and 3 V thresholds were only numbers for the example above. See the datasheet for the real thresholds, from which you can compute the real high and low times, and from that the period and frequency. Also note that 100 Ω is a excessive load for most real inverters.
Step 1. To get 15V, you want a power supply higher than 12V. Well, the NE555 has an output which can drive about 100 ma. Use this to power a voltage doubler circuit (2 diodes, 2 capacitors) and you'll get a DC supply slightly less than 24V. Searching for "NE555 voltage doubler" should cover the details.
Step 2. Use this supply to power a low-current opamp with a gain of 2 driving the varicaps,
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If you mean gradually change triangle form to sawtooth, with same amplitude/frequency, then I think a variation on the classic op-amp triangle wave circuit can be a starting point, but it distorts the frequency. You could replace R2 with a bridge and two current sources?
Here I_1 controls amplitude, V_IN controls level.
For a symmetrical wave, V_MID is mid supply, but you can vary V_MID to change ratio of rise rate to fall rate. Here's a falstad simulation to play with.