The price of the bike is not too much of an issue because I will claim it on my Self Assessment
From what you've said, HMRC won't allow you to claim the whole purchase cost. To do that you'd need to prove that the bike is used wholly and exclusively to conduct your trade - you'd not be allowed to use the bike for any personal mileage AT ALL.
That might be (IS) very difficult to prove, especially if it's your only form of transport. As you've said:
...but I do not drive and will be interested in maybe using the bicycle to travel long-ish distances (20 to 40 miles) to visit friends and run errands of my own when not working for Deliveroo.
IIRC, all you can claim is 20p per mile, and you need to keep track of your "working" mileage (as you do if using a personal car for work).
And you can't even claim the mileage for travelling elsewhere to start your shift - that's classed as personal mileage:
Had I travelled to another town on the bike before starting my shift, those would probably be a lot higher...
At 20p per mile, you'll need to cycle 20,000 delivery miles to recover the cost of a £4000 e-bike.
You maybe want to check this with your own accountant/book-keeper before spending so much on an e-bike.