Explicit calculus of Hilbert-Smith conjecture

Here is an answer in the setting of compact (boundaryless) 3-manifolds. Note that if $H$ is a compact topological group acting continuously on a compact metric space $M$, then there is a $H$-invariant, necessarily complete, metric on $M$. Thus, the existence of an invariant metric is not a restriction.

First, consider actions of finite groups $H$. Even if you assume that the action is free, there is virtually nothing of interest that can be said about manifolds admitting such actions:

Suppose that $M$ is a compact connected 3-manifold which is not $S^3$. Then $M$ has a nontrivial regular covering $p: M'\to M$ of finite degree. The deck-transformation group of $p$ will act freely and effectively on $M'$. If $M$ has infinite fundamental group then there is an infinite tower of such coverings. This existence result is nontrivial, we know it only due to the proof of Thurston's Geometrization Conjecture. For instance, given any finite group $H$, there is a compact 3-manifold on which $H$ acts freely (and there is enormous supply of such manifolds, both geometric and nongeometric). There is no hope classifying such actions. The best one can do is:

Suppose that $H$ is a finite group acting smoothly and effectively on a compact geometrizable manifold $M$. Then there is a geometric structure on $M$ (in Thurston's sense) invariant under the action of $H$.

I can find the precise reference if you are interested, but it is a very hard result, a culmination of work of many people. Even the case of finite cyclic group actions on $S^3$ was unknown for about 40 years and called the Smith Conjecture.

Things get only worse if we allow non-free topological actions. For instance, $S^3$ admits an involution whose fixed-point set is a wild sphere (this is due to Bing). This action of $Z_2$ on $S^3$ cannot possibly come from a subgroup of a Lie group acting smoothly on $S^3$.

I will, therefore, assume that the compact Lie group $H$ has positive dimension. Let $H^0$ denote the identity component of $H$. Thus, we have a nontrivial connected compact Lie group $H^0$ acting effectively on our manifold $M$. In what follows, I will restrict my attention to such actions, so $H=H^0$. These were classified in 1960s in the work of Neumann, Orlik and Raymond. Their papers appeared in

Mostert, Paul S. (ed.), Proceedings of the conference on transformation groups, New Orleans, 1967, Berlin-Heidelberg-New York: Springer-Verlag. xii, 456 p. (1968). ZBL0167.00104.

Note that a connected compact Lie group always contains $S^1=SO(2)$. The hardest part of classification is to classify such actions. This was done in

Orlik, Peter; Raymond, Frank, Actions of SO(2) on 3-manifolds, Proc. Conf. Transform. Groups, New Orleans 1967, 297-318 (1968). ZBL0172.25402.

For simplicity, I will assume that our manifold is orientable (see their paper for details on the nonorientable case). Then, if $M$ is a compact 3-manifold which admits an effective action of $SO(2)$, then:

a. The case of an $S^1$-action fixing at least one point. Then $M$ is a connected sum where each summand is either $S^2\times S^1$ or a lens space (one can have a mixture of both). Moreover, the converse is also true: Every such connected sum admits an effective $S^1$-action with nonempty fixed-point set.

With few exceptions, these manifolds are non-geometrizable: The only (geometrizable) exceptions are $S^2\times S^1$, lens spaces and $P^3 \# P^3$.

b. Suppose that the action does not have a fixed point. Then $M$ is a Seifert manifold, hence, is geometrizable, and the action comes from the Seifert fibration (which is unique apart from a few exceptions). Seifert manifolds are geometric and one can show by case-by-case analysis that the $S^1$-action comes from a geometric structure in Thurston's case, but you may have to switch from compact group to a noncompact group. (The quoted papers were written before Thurston's work, so they did not use the geometric language.) For instance, if $M$ is an $H^2\times R$-manifold, then the circle action comes from the action of the $R$-factor acting by translations. (Lifting the circle action to the universal cover, one gets an $R$-action.)

From the list given by Orlik and Raymond, you can see that $T^3\# T^3$ does not admit an effective circle group action, hence, no action of a compact Lie group of positive dimension.

The actions of compact groups on 3-manifolds whose principal orbits are 2-dimensional were analyzed by Neumann (completing and correcting the earlier work by Mostert) in

Neumann, W. D., 3-dimensional G-manifolds with 2-dimensional orbits, Proc. Conf. Transform. Groups, New Orleans 1967, 220-222 (1968). ZBL0177.52101.

The class of manifolds is even more restrictive: These are lens spaces, $T^3$, $S^2\times S^1$, $P^3\# P^3$. (There are few more non-orientable examples.) Neumann also classifies the actions themselves. In particular, all these manifolds are geometric. From what I saw, my impression is that, for some choice of a locally homogeneous metric from Thurston's list all these actions come from actions of subgroups of the full isometry group of that metric. But it's worth verifying this.

Lastly, one should consider actions with 3-dimensional principal orbits. I did not chase down the references; since you are interested in the question, you should read the references given in the papers above to find out. I think, the only manifolds admitting such actions are $T^3$ and some spherical space-forms.