>For example, I live in London and enjoy this city. I would love to be able to stay at home and work on my side projects in hope of making extra money, but given the outcome I believe from this, everyone who was to only accept the basic income amount would no longer be able to afford to live in London and would have to relocate to less desirable places in the country.
Some models call for a universal income which takes location inconsideration. The income would be higher in London, for example. That's a pretty controversial model, though.
That's obviously nonsensical. All the basic income would be sucked up by rising rents, raising cost of living, go to step 1. Basic income should be set up to pressure people to live where it is cheaper.
In any monopoly/duoploy/oligopoly/cartel-ish environment in which prices are determined by limited supply and therefore buyers ability to pay (seller gets the economic profit) makes buyer side cash infusion useless.
If we assume people prefer to live in a major city, and incomes are adjusted so that your income matches your cost of living, there is no downside to living there. That puts upward pressure on an area that is probably already maxed out for living space, which causes people to pay a little bit more in order to secure a place to live. Then, your income rises to offset your new costs of living there, providing the money necessary to allow someone to pay just a little bit more than you, thus beginning the cycle.
Outside of the city you would find a low demand and high supply of housing under such a scenario, and thus there would be no fuel to ignite a race to out-pay each other to find a place to rent. In fact, there you could probably convince the owner to accept a little bit less for rent as few people are interested in the place at all. If your offer is not accepted, it might sit vacant.
Note that income can be adjusted even if it doesn't exactly match the cost of living. For example, if living in the capital is 25% more expensive basic income could be 15% more in the capital. That might allow a lot of people to live in the capital with basic income. I don't know how that affects your scenario.
Some models call for a universal income which takes location inconsideration. The income would be higher in London, for example. That's a pretty controversial model, though.