Fast (high) frequency hopping with off-the-shelf components

I would concentrate on the VCO and try to find a design that has a reliable change in frequency per volt. Something that can almost be run "open-loop".

This is a priority because my suggestion would be to use a PLL but with feedforward to get you the speed (and distance to the next hop) then the PLL loop would tweak the final few MHz. So how accurate and in what time do you want it to be?

You can get quadrature outputs using two resistors, an inductor and a capacitor across a vast range of frequencies but with some amplitude variations.

One option would be DDS in the lower frequency range (e.g. AD9956) and double conversion with a LO switchable in 200MHz or 400MHz steps.

While double conversion may look more complex, it means you can avoid I-Q mixing, because it allows you to place the image well out of the band of interest.

For example, if DDS can comfortably cover 100-300MHz, then a simple conversion with a LO of 500MHz would give an upper sideband of 600-800MHz and an easily rejected lower sideband.

Then another simple conversion with four spot frequencies of 1.0, 1.2, 1.4 or 1.6GHz would cover the required range. (The last LO reaches the bottom of the band here, so the mixer should be balanced for low LO leakage)

Variations are obviously possible; if DDS can easily cover a 300MHz low band range, then only 3 spot frequencies would be required. It may be possible to make the 2nd LO spot frequencies harmonics of the first LO. And so on.

One option is using a DDS that can reach 1.2 GHz, with a frequency doubler.

Doublers are essentially just some nonlinear circuit to produce harmonics with some filtering to pick out the preferred 2nd harmonic at the output, so they don't require any lock-in time when changing frequency (aside from that implied by the bandwidth of the selection filter).

Doublers tend to allow at least a bit of the input frequency (perhaps 20 dB or so below the 2nd-harmonic output), and also its 3rd harmonic, through to the output, so some careful filtering, or even an adjustable filter, might be required if you need a very pure output frequency.

Doublers also tend to be a bit fussy about the power level at the input, and produce an output attenuated from the input level, so you may need some additional amplification and/or attenuation to get the scheme working well.

Two application circuits are given in the [AD9566] datasheet... What would be the switching time from, say 1.6 to 2.4 GHz for these?

Those are both essentially PLL schemes. The switching time will be limited by the bandwidth of the loop filter. I'd expect it to be difficult to get it below a several 10's of ns. Although 50 ns doesn't seem totally out of the question if the loop bandwidth can be as high as 20 MHz. (This also applies to your proposal of a straight PLL solution)