You're going to have to get under the beast with a few clean rags and clean it off, to see where it's all coming from.
The oil/coolant heat exhanger *could* be a contender. It's made of alloy and one of the coolant pipes to it make have fractured, letting the coolant out. It would be unusual though (they don't corrode and they don't get a stressed life) and any failure of the coolant pipes would not leak oil.. just coolant.
My money is on "it's the radiator"... these live in the direct line of road-salt/grit and because they're big and flat, they can twist and suffer from vibration. A split/burst at the bottom left or right hand corner might also blow coolant onto the oil/water heat exchanger (I don't know the TA.. but the heat exchanger is usually behind the rad) so it might just have the appearance that it's the heat exchanger.
Radiators are not too dear and are simple to fix. Heat exchangers are also not too tricksy .. but may cost a bit more (since only Fiat will tend to have them)... but I don't know the exact prices.
Oil on the sump is probably unrelated. Sumps can be oily if the sump is leaking a bit. They're just sealed with some silicone paste (like around your bath) so a leak is "normal" and not serious if it's just "oily".
Most people just leave this alone, since an oily sump stops the sump pan from corroding... but if it's dripping/leaving stains on the floor then it needs to be re-sealed.. but it's still not serious so no rush.
Obviously, if your mum's car is dropping oil all over the floor, then this is not an "oily sump" problem... but you would need to find out where it is coming from, if it's not part of a heat exchanger problem.
Ralf S.