Two major systems dominate the map: a rather intense Atlantic low arriving over Ireland in the morning, and a occluding low over Russia. The latter has instability zones along a frontal system stretching from southern Romania to Estonia and eastward into Russia.
At the start of Tuesday, a moist maritime airmass is advected from low latitudes via Portugal northward into Ireland and western UK, where elevated thunderstorms are likely to occur. The cyclogenesis strengthens the wind field and bends the plume eastwards into France and the Benelux, meeting another moist plume of SW Mediterranean air at about 18Z over northern France. The plume of unstable air will reach western Denmark during the next morning hours.
A complex low level flow pattern with lower absolute humidity but unstable stratification is present over the Iberian Peninsula. South and east of Spain GFS predicts accumulation of moister air (>12 g/kg) which gets involved with cyclogenesis pulling in dry air from the Sahara. This development may initiate storms as well.
...N France, Belgium, Netherlands, W Germany, W Denmark...
From 06Z to 15Z, only marginal, spotty instability release is anticipated by GFS (shallow uncapped parcel layer depth) in the northwestern half of France and the Channel/southern North Sea region. Only after 18Z, and in the current GFS 18Z run around 21Z, significant deep convergence and CAPE release will be possible from a deeper layer elongated region from north-central France into the North Sea. Most quasigeostrophic lifting occurs over the Channel and UK.
CAPE itself will be nothing special, but the environment of 20 m/s 0-6 km shear vector length, >10 m/s 0-1 km shear and some 100-200 m2/s2 SREH over 0-3 km allow multicells transforming into an MCS, probably a (squall) line, as there is a 10 m/s 0-2 km shear component normal to the moisture axis.
The MCS may reach maturity after 00Z/03Z and pose a severe gusts threat (although higher LCL heights would be more favorable, but there is a rather strong steering flow). Rotating updrafts may initially be possible with some isolated large hail, and a brief tornado cannot be excluded on the basis of sufficient low level shear and very low LCL heights.
Note that in case of a squall line it may eventually go faster east/northeastwards than indicated by a large scale model like GFS.
...Spain, SW France...
Last two model runs have consistently suggested that storms will initiate in a convergence zone between central and northern Spain during the afternoon. These storms are likely to drift off / regenerate in the direction of southwestern France towards / into the evening, clustering into an MCS. While there is moderate deep layer shear, more than an isolated large hail event during the afternoon is not likely.
The rest of the Iberian region looks messy with respect to the instability and triggering factors. Instability increases towards the south, moderate deep layer shear and some helicity are present, and can yield large hail, or an isolated severe gust given that the air is also drier.
The best shear and SREH >300 m2/s2 are found along the east coast, where the 18Z GFS shows the passage of a weak shortwave trough during the day. But diverse parameters suggest instability is mostly elevated. For example, 0-1 km MLCAPE is very low, ICAPE is high but ICIN too (column-integrated CAPE and CIN), indicating either capping or elevated conditions. In principle the shear seems adequate for supercells, buth confidence is low.
The shortwave may rather produce a lot of rain over a large area.