Battery - Need to have easy swap battery packs and charge them at petrol stations. This is the only sensible route because 1) nobody wants a car where the £10k battery needs replacing after 3 years and 2) It's the only way to 're-fuel' quickly. Weight is not an issue, a standardised design and mechanical assistance could even make it automatic.
I think weight is an issue, given that EV battery packs can weigh 500Kg. That's a lot to be moving around if EVs replace ICEVs, especially as battery stations would see several times as much use as petrol stations (even with a 500Kg battery, range is a little over 200 miles). Size is also an issue. It's not a deal-breaker, but it is an issue with the current size and weight of batteries and the very large number of swaps that would have to be done.
Huge improvements look imminent, though, possibly an entire order of magnitude. Not just on paper - it's already well into the prototyping stage. A 50Kg battery the size of a small suitcase would be much easier to handle for battery swapping.
The rate of improvement shows up a practical problem, though. How can you standardise the design when the technology is changing so much? You run the risk of either stifling development by locking it to the standard or making the standard rapidly obsolete. The possibility of rapid obsolescence would make it difficult to get the standard accepted at all - what company is going to pay for the infrastructure that might be obsolete in a few years?
The cost of the battery will have to be met by the driver in one way or another. You won't have to pay £10K every 3 years for a new one, but you will have to pay extra on each swap to cover the cost of the depreciation of the battery. Although the next generation of batteries should lose maximum charge much less quickly, so the cost would be much less.
Hydrogen - Need to build expensive tanks to store the Hydrogen at petrol stations, tankers no doubt more expensive too.
Much more dangerous as well, due to the very high pressure. The ongoing expense would be higher too, because whatever you store it in will break much quicker than storage for petrol or diesel. Hydrogen is a real bugger to store.
But the biggest problem is getting the hydrogen in the first place. I'll belabour this point again, because it's the most important one. There is no known way, not even in theory, to obtain hydrogen on a large scale without a huge waste of energy. Hydrogen is always (on Earth) found bound to other atoms. To get just hydrogen, you need to break those atomic bonds. That requires energy - more energy than you get back from passing the hydrogen through a fuel cell. Hydrogen is not an energy source. It's an energy carrier and it's a very inefficient one.
It's possible that the efficiency of hydrogen as an energy carrier might be improved to possibly usable levels with odd approaches such as using biological reactors based on termite guts (seriously, it is being investigated), but they are only possibilities in early stages and even if they exceed the best hopes they still won't make hydrogen a fuel. Just a less inefficient energy carrier.
What is never going to happen is re-fuelling at home, apart from a few top-up chargers for people who have drive ways we are not going to see millions of people relying on the 13a socket to provide for ther daily motoring needs.
True. There will be some use for home charging using much higher power than a 13a socket, but it will only be viable for some of the people who have driveways or garages.
Once you get away from that fiction you can work on practical solutions.
Which, thankfully, some people are doing.
Of course, you then have the problem of supplying the energy. Moving away from oil as a fuel doesn't mean reducing the amount of energy required. It just changes the form of the energy. A very large increase in electricity generating capacity would be required. That's not an insurmountable problem, but it is a problem.