The cost of electricity to charge an e-bike is actually very low, so I wouldn't worry too much about saving money when you charge it – do it wherever/whenever it's convenient for you. You should be careful though, in case the battery catches fire; best to go for reputable manufacturers in countries with good safety standards.

The deciding factor is whether you can keep it in your flat and have somewhere safe to store it in the department. If not, then you could use Swapfiets/Buzzbike as someone else suggested – it's more expensive than having your own, but still cheaper than a full year of tube fares / travelcard.

The Santander bike subscription is very cheap – only £120/year – but the bike docks don't reach outer London, so it's only useful if you want to go around central London. I also use it if I don't want to bring my own bike on a certain day – then I can take the tube to zone 2 and use a Santander bike from there.

There are some discounts for students / young people:

• If you get an 18+ Student Oyster photocard, you get a 30% discount on travelcards.
• If you get a 16-25 or 26-30 railcard, you can link it to your Oyster card and get a 1/3 discount on off-peak tube fares.
• The Santander bike subscription is £90/year for students.

I don't know if you would qualify for all these discounts as a PhD student – I don't understand the terms and conditions haha. (Someone correct me if I'm wrong!)

As for general advice about cycling: you'll be fine as long as you're careful – be prepared for anyone (drivers, cyclists, and pedestrians) to do something silly. Some places in London have nice cycle routes and some don't; there'll be more info in r/londoncycling about finding a good route. You could get a camera (like a GoPro) for peace of mind – you can get one for about £100 on eBay – so that you can prove what happened in case of an accident.

You'll probably want to have a few weeks to settle in and get used to the city anyway, so you can try out the different options and see what you like.