See this MO answer of mine: mathoverflow.net/a/155122/11211 That being said, none of these references really are any easier than Glimm-Jaffe, they complement and help the latter but don't actually circumvent it. Constructive QFT is a difficult and highly technical subject with lots of open problems, and the the available textbooks on the subject just reflect that state of affairs.

The best short intro to nuclear locally convex (vector) spaces (lcs) is the little book by A. Pietsch, "Nuclear Locally Convex Spaces". Pietsch was the pioneer of the approach to nuclear lcs using operator ideals instead of modelling the former on the theory of topological tensor products as Grothendieck originally did in his PhD thesis. That being said, this is a very terse book which still requires previous knowledge of functional analysis and maybe Pietsch's approach to nuclear lcs isn't the one you seek... Btw, Glimm-Jaffe's book on constructive QFT also briefly discusses nuclear lcs.

@Isaac we may continue this discussion by email, no problem, also because we're kind of departing from the strict matter of the OP.

Generally there are many ways to continuously extend a distribution $u$ from $\mathscr{D}(U)$ to $\mathscr{D}(\mathbb{R}^n)$, taking boundary values of $u$ identified with a real analytic function $f$ on $U$, which then admits a unique extension to a holomorphic function on an open domain in $\mathbb{C}^n$ containing $U$, is a particular case. These extensions are collectively called renormalizations, and their existence is ensured by the Hahn-Banach theorem.

I should also qualify my 1st comment: "boundary values of $f$" are really meant in the sense of holomorphic functions viewed as distributions. The imaginary part of their argument is seen as parameters sent to zero in the distribution sense while keeping the real part at the complement of $U$ - depending on the direction one approaches zero, one may get a different boundary value.

The first paper by Osterwalder and Schrader on the reconstruction of the Wightman $n$-point functions from the corresponding Schwinger $n$-point functions overlooked that point, and a proper discussion of this matter in the context of hyperfunction theory was provided in their second paper (as well as in an independent paper by Glaser).

In due time: as Abdelmalek said at the end of its answer, strictly speaking it may not always be the case that if one starts with an analytic function $f$ on $U$ as in the end of 3. above, the boundary values of $f$ on the big diagonal exist in the tempered distribution sense. One needs polynomial bounds such as the one in the OP to ensure that, otherwise one generally only obtains that way what is called a "hyperfunction", which is even more singular than a distribution.

The perturbative QCD S-matrix (which, as you correctly pointed out, involves an infinite-time limit as well), on its turn, is physically reliable only in the high-energy limit, thanks to asymptotic freedom.

The point is what one is able to compute (or prove the existence thereof) for a given QFT model. Lattice QCD calculations assume both finite volume (i.e. infrared) and short-distance (i.e. ultraviolet) cutoffs, so there's no renormalization involved. We do know by the work by Balaban and Magnen, Rivasseau and Sénéor using weak-* compactness sorcery that the continuum limit of pure Yang-Mills Schwinger functions exist for finite volume, but we don't know anything about uniqueness, let alone its thermodynamical limit. For full, matter-coupled QCD we don't even have that.

For instance, the perturbative series for quantum chromodynamics in the low-energy regime (e.g. nuclear physics) doesn't seem to be asymptotic at all, due to quark confinement. Color charged states simply don't appear in scattering at that regime.