Why does falling global bond yields signal coming deflation? This article suggests that, "Bond markets are signalling something very nasty coming down the road at us – an all encompassing, worldwide deflation. "

I cannot understand why bond yields falling signals deflation. Is it because falling bond yields indicates a weakening global economy? Or is the transmission mechanism more direct?
Bond yields falling from their current near-zero position will place them in negative yield territory. Negative bond yields are deflationary by definition.
Paragraph 3, sentence 5 of the article says:

With Bank Rate already close to the floor, and some UK bond yields now in negative territory... [emphasis added]

To understand why negative bond yields are deflation markers, consider the example of negative Asian and European central bank rates described in the same article:

This is because a negative rate is effectively just a tax on the banks, forcing them to pay for the privilege of holding reserves with the central bank.
Someone has to shoulder the burden of this tax; either borrowers pay more, or depositors get less, or profits go down, and if it is the latter, then it damages the banking sector’s ability to rebuild capital, crimping credit availability accordingly. Whichever it is, the end result is a monetary tightening, rather than the loosening intended.

So you can think of the effect of negative central bank rates (and, similarly, negative bond yields) as having the inverse impact as quantitative easing which creates an inflationary effect.
This page explains the relationship between prices and money supply as follows:
$$P=\frac{MV}{Y}$$
where $P$ represents prices, $M$ is the money supply, $V$ is the velocity of money and $Y$ is the real GDP.
Therefore,
$$\Delta P = \Delta M \cdot \frac{V}{Y}$$
and for constant $V$ and $Y$,
$$\frac{\partial P}{\partial M} = \frac{V}{Y} &gt; 0$$
So an effective restricting of the money supply caused by negative rates/yields via the mechanism above described will tend to put downward pressure on prices (a/k/a deflation).
Edit:
In the comments to this answer, OP asked about the relationship between negative interest rates and negative bond yields.
Whereas central banks set rates by policy, long term bond yields are determined by the market (i.e., supply-demand equilibrium). An inverted yield curve signals recession and implies rates are set too high relative to market expectations. If central banks set rates in positive territory when long term bond yields are negative, they will de facto force the yield curve to invert. Although this sometimes happens, it is usually counterproductive to the policy goals of the central banks (in most environments) and, therefore, typically avoided.