Your maximum will be the capacity of 6mm2 wire
or probably 32A.
"Wait! That's the maximum the Type 2/Mennekes/J3068 standard supports, and is ludicrously fast at 22 kW! How can an old house support that?"
Because your demanding requirements force us into very particular advanced tech which is capable of that. It's going to cost more and be tricky to acquire, but you have precluded all other possibilities. And since we have the technology, we might as well use it to full effect.
Your service is fairly modest, and your charging plans extremely ambitious. Also, you are unwilling to fund running cabling from the main service panel/consumer unit to the garage. That forces us to use the existing feeders, which fortunately are 6mm2 and 32A even under the most conservative rules. I don't know why they are 20 amp breakers - this is something you can address with the electrician to open up that pathway to at least 32A. Whatever that is, that constitutes your charging limit.
The trick being the other loads in the house
And this is why we are forced into a particular technology - EVEMS. This places "CT" current meters on the main supply wires from the utility. It is monitoring the current draw of all loads in the house including EV charging. It then communicates with the "charger" (which is not a charger at all) which then adjusts charge current dynamically to prevent the house's electric service from overloading.
Normally I would link Technology Connections' video about letting go of range anxiety and install a more modest system, but given the limitations of your house, I think we are forced into this technology. So we might as well make the most of it. Note that high-power EV charging is by far the most demanding load a house will ever face. So it is also the most unforgiving to even the slightest flaw in an electrical installation. You have been warned.
Because of your "no cables" prohibition, we are forced into the Myenergi ZAPPI for the charging station, which calls this feature "Grid Limit". This can partner with a power meter called the HARVI, which can communicate with the ZAPPI via radio link (solving your "cannot run a wire" problem).
Mind you, I have not kept up on the latest European product selection. It's possible someone else has an EVEMS capable station with radio link to its power meter.
The grid limit will need to be set to the MINIMA of your electric service from the utility, the capacity rating of every wire and circuit breaker and equipment enroute.
Most of your charging happens at night, when loads in the house are quiescent, so you will get full charging power. At other times, you can increase EV charge rate by turning off the other loads in your house.
Phase balancing will help
If you followed the link I provided to Technology Connections' "What an EVSE is" video, Alec was incorrect that Europe doesn't use that. They certainly do use exactly the same technology, differing only in 2 additional phase wires being added. Indeed, American cars (e.g. brought over for US soldiers) will plug right into Type 2 Mennekes either with a Type 1 tether cord, or an adapter made of metal and plastic.
But here's the point: There are 3 phases, but only one CP wire indicating authorized amp draw. This means that in the 3-phase Mennekes system with EVEMS, the signal must conform to the heaviest loaded phase. If you have 32A electric supply, wish to deliver 32A to the car, and your three phase loadings are 0A / 2A / 21A, such that they have 32A / 30A / 11A of headroom, regrettably EVEMS must tell the car "you may only take 11A".
Thus, you should take care to see that single-phase loads likely to be used together are on different phases, so that in that same case, the three phase loadings are 8A / 7A / 6A for instance, providing 24A of headroom.
