Changing routes/strategies for delivery vehicles might not be specifically named as an aim of Lambeth's LTNs but at a city wide level, introduction of LTNs along with other things is part of a general policy to reduce overall capacity on the road network. If it didn't reduce overall capacity then no-one could make the argument that in the longer term, it reduces the overall amount of traffic, including on main roads. That's traffic evaporation and without it, then of course the main roads would simply become loads busier with displaced traffic. Some people deny that this happens, but the arguments have been gone over a million times already on this thread.
If you change the overall capacity and also change with roads are through roads, then of course delivery companies will have to make some changes to the way they do things.
While I can see that this may well cause some short-medium term pain for delivery drivers while the system re-organises itself, I don't think it's true to say that the burden is simply shifted to the drivers in the longer term. The delivery companies already pay drivers (I'd assume) pretty much the lowest they can get away with. That means that if the amount a driver can make in a day reduces much further, the companies will start to find that drivers no longer want to work for them. The incentive will be for the delivery companies to adjust things so that on average drivers can make a similar sort of amount of money per day. How they do that exactly I don't know but I'd imagine it would be a mixture of changing what areas are allocated to what drivers, and maybe increasing the pay per drop if they have to. Perhaps changing the mix of vehicles they use to include some that can go through LTN gates.
If anyone ends up taking on the consequences of these changes it'll be the end customer. Delivery/collection charges might go up a little. Perhaps pricing will be changed to encourage more picking up at local hubs like local shops. That's all fine with me; I would say that delivery charges are probably too low as it is, making it too easy for people (including me) to order loads of stuff to their door which has consequences for the amount of traffic on the roads generally.
Yes, all this amounts to quite a bit of disruption and the transition won't necessarily be easy for delivery drivers. Some jobs might disappear and new ones might appear. The Pedal Me thing has already been mentioned. I would say that in any case, without any LTNs being involved, the pandemic by itself would have caused all sorts of changes in the delivery industry. Many things are in flux just now and we'll not know exactly how things are going to shake out until some time has passed.
I talk to one of my regular Hermes drivers when she delivers stuff. A few months ago she told me she was thinking of giving up, because of the amount of workload they were pushing on her. When I last saw her, last week, she was still doing it and was telling me that Hermes have been bought out, and she was waiting to see if this was going to result in changes in working conditions. It seemed like she was hopeful it might make things better; I've no idea if that's realistic. In any case, lots of things are changing right now. For example I've been following the story of a company that is going to start transporting parcels by rail, with trains being loaded/unloaded at stations and forwarded on by small vehicles. This has the potential to allow delivery companies to get things moved in bulk into city-centre locations without putting lorries on the roads - rather than relying on distribution centres located on the periphery and then multiple road vehicles driving into the city. In fact this (if successful) is in some way a return to how things were delivered before we shifted nearly all of our goods transportation to road-based systems. Approaches such as this one are encouraged by making it less convenient for delivery companies to do everything by lorry and van. I would see it as a reversal of changes that were made in the mid 20th century that turned so much of our urban areas into traffic dominated disasters, with all the consequent things like air pollution.