  With the rapid advance of wide-field surveys it is increasingly important to
perform combined cosmological probe analyses. We present a new pipeline for
simulation-based multi-probe analyses, which combines tomographic large-scale
structure (LSS) probes (weak lensing and galaxy clustering) with cosmic
microwave background (CMB) primary and lensing data. These are combined at the
$C_\ell$-level, yielding 12 distinct auto- and cross-correlations. The pipeline
is based on $\texttt{UFalconv2}$, a framework to generate fast, self-consistent
map-level realizations of cosmological probes from input lightcones, which is
applied to the $\texttt{CosmoGridV1}$ N-body simulation suite. It includes a
non-Gaussian simulation-based covariance for the LSS tracers, several data
compression schemes, and a neural network emulator for accelerated theoretical
predictions. We validate our framework, apply it to a simulated $12\times2$pt
tomographic analysis of KiDS, BOSS, and $\textit{Planck}$, and forecast
constraints for a $\Lambda$CDM model with a variable neutrino mass. We find
that, while the neutrino mass constraints are driven by the CMB data, the
addition of LSS data helps to break degeneracies and improves the constraint by
up to 35%. For a fiducial $M_\nu=0.15\mathrm{eV}$, a full combination of the
above CMB+LSS data would enable a $3\sigma$ constraint on the neutrino mass. We
explore data compression schemes and find that MOPED outperforms PCA. We also
study the impact of an internal lensing tension in the CMB data, parametrized
by $A_L$, on the neutrino mass constraint, finding that the addition of LSS to
CMB data including all cross-correlations is able to mitigate the impact of
this systematic. $\texttt{UFalconv2}$ and a MOPED compressed $\textit{Planck}$
CMB primary + CMB lensing likelihood are made publicly available. [abridged]
