Part of me thinks this is too broad while part of me doesn't. See, there are so many different physical constants in the universe: $c$, $\sigma$, $\alpha$, $\epsilon$ . . . You get the idea. Change any of them and something odd will happen. Mess up the fine-tuning and something bad will happen. And that's the gist of the whole thing: Change something very small - such as the mass of the top quark - and things can get very messy very quickly.
So I could choose to analyze any or all of these constants, and I would rapidly run out of room, especially given my inability to be concise. But I've done a bit of reading, and I think I know how I can narrow it down (for some interesting papers, none of which I've gotten through much of yet, see Coleman (1977)1 and Matusmoto et. al. (2010); see also this answer and this question on Physics.SE, and links therein).
The most important (in this case) paper I found was Coleman & De Luccia (1980). It investigates the effects of gravity upon a false vacuum. It seems trivial until you get to this part:
At first glance, this seems a pointless exercise. In any conceivable application, vacuum decay takes place on scales at which gravitational effects are utterly negligible. This is a valid point if we are talking about the formation of the bubble, but not if we are talking about its subsequent growth.
So what governs the global evolution of the bubble? Gravity. Let me see what I can work through to show the point.
Take a scalar field, $\phi$. There is energy associated with any location in the field. To analyze this field, we can take its action, $S$. In this case, $S$ depends on the Lagrangian, $L$. Using Coleman and De Luccia's notation,
$$L=\frac{1}{2} (\partial_{\mu} \phi)^2-U(\phi) \tag{1}$$
Here, we use the convention where $\partial_x$ denotes taking the partial derivative with respect to $x$. This is a simple Lagrangian, like a particle moving in a gravitational field (with very small changes in $y$, because then $g$ would vary greatly):
$$L= \frac{1}{2}mv^2-mgy$$
So we can take the action using $(1)$:
$$S = \int d^4x \left(\frac{1}{2} (\partial_{\mu} \phi)^2-U(\phi)\right) \tag{2}$$
The defining feature of $\phi$ is that it has two relative minima. These correspond to a false vacuum and a real vacuum - although the "real" vacuum could also be a false vacuum, simply at a lower energy state.
The thing is, if we consider gravity, then we insert a few additional terms into the action. It becomes
$$S = \int d^4x \sqrt{-g} \left(\frac{1}{2}g^{\mu \nu} \partial_{\mu} \phi \partial_{\nu} \phi - U(\phi) - \frac{R}{16 \pi G} \right) \tag{3}$$
If you're familiar with the basics of general relativity, then this should remind you of the Einstein-Hilbert action:
$$S_{\text{EH}} = \int d^4x \sqrt{-g} \frac{R}{16 \pi G}$$
So our Lagrangian is really similar to a mash-up of those two. Well, not really. But close.
The point is, by modifying this action, we've introduced an additional term (well, we've done more than that). We've put in a new cosmological constant. This is crucial. We can figure out the equations of motion without making the modifications and then with modifications. The difference is that the bubble can grow or shrink in different ways.
Sections II and III aren't important at the moment, and as the authors write,
We have tried to write it in such a way that it will be intelligible to a reader who has skipped the intervening sections.
Taking that to heart, we can go to Section IV.
Take a hyperboloid defined by an expression of $\Lambda$. Starting from the metric
$$ds^2 = d \tau^2- \rho (i \tau)^2 (d \Omega)^2$$
where $d \Omega$ is hyperbolic, Coleman and De Luccia show that an FLRW metric can be constructed for the universe. I won't go through the derivation, because it's not conceptually important, but the metric is
$$ds^2 = d \tau^2 - \Lambda^2 \sin^2 (\tau / \Lambda)d \Omega^2 \tag{4}$$
This universe will either expand or contract. Looking at earlier relations given, it is clear that this is influenced by, among other things, $G$.
Gravity doesn't influence a lot of the processes in the false vacuum, but it influences some.
For some information on what things would be like in a normal universe with a small change in $G$, see my answer here. Other great answers to a slightly different question can be found here. In summary, the most important effect of a change in a constant would be a change in $G$, which could stabilize the false vacuum and influence (a little) its growth rate.
1 Coleman and De Luccia originally investigated false vacuum scenarios, though Coleman is considered by some to have priority.
