Ah it appears the Kenwood suggestion was to help negotiation, not actually to instal it.Well that's the modern way of doing it. You're basically attempting to keep the moisture out. When concrete floors are involved its a bit difficult to comment without seeing it. Where in London is this Onion?
You'll get a 100 different opinions on this. Personally I think a liquid DPC is probably sound technology but may not be as effective in the wonky, imperfect brick work of certain old houses. The people who sell it however don't really understand how old houses work and tend to prey on the gullible and elderly who occupy them. Tanking is far less of a great idea imho, it will never allow the wall to dry properly (has to breathe both ways) and will probably only ever serve to hide a failed liquid DPC.
Kenwood are well known round these parts for being on the pricey side so I would try and get a few more quotes if you go down this route.
It only has cement render at the back (and around extension)- the front is the original brick apart from a little render at the bottom which would need removing.Thanks for adding photos to the opening post. Certainly cement and damp injections will make things worse if anything. Water will always make it's way in from the ground and the wall will get wetter and wetter if it can't dry out and it will find a way out somewhere.
From the stains it does look like there is a bit of damp in the wall. That's going to be annoying so worth thinking about how you are going to fix it..
The ideal Victorian terrace would have had lime mortared brick that evaporated any excess water away to the outside. Inside would have been lime plastered so water from the inside could soak in rather than cause condensation mould problems.
Yours looks like it has been cement rendered outside , and fairly recently judging by the lack of dirt. It will likely ruin the bricks if you try to get that off, so unless you get lucky you would need to remove the cement and lime render if you want the wall to be able to evaporate damp to the outside. That's expensive. Groundworks to lower the outside to a height lower than the inside will be expensive. Those are the two damp fixes that tend to work best.
You might get away with improving underfloor ventilation where you have a suspended floor. There will be bricks that can still breathe down there and getting the water away before it goes up the wall is possible with decent ventilation.
It's a stressful time buying houses. I would be much more comfortable buying a house that still had the original bricks and lime mortar exposed and the original windows. If they can last 100+ years without any real intervention then maintenance is cheap. The modern system of intervention every few years only benefits the building trade.