It depends somewhat on which city you're talking about, but this sounds very similar to the business model of Uber. It almost sounds like you're considering becoming an Uber driver, with the exception that you're making trip destinations match their origin. Uber is active in cities like London, Birmingham, Bristol, Manchester, Glasgow, etc. and has a sizeable base of both riders and drivers, indicating a regulatory environment that is favorable or at least not sufficiently hostile to such a model.
London has regulations restricting use of taximeters, but last fall the high court ruled that Uber's app does not constitute a taximeter that would violate the regulations. The mayor said this was the primary issue around the legality of the service there, and this is the main regulatory dispute in Wikipedia's list (which may not be all inclusive). January headlines from this year heralded the lack of regulatory obstacles to Uber's operations in that city.
You should probably first check with your insurance company, especially if there are FAQs on their website or details in your insurance policy, because you might not be covered under that policy on those drives without some additional premium.
Check out the UK's Licence Finder to help find out what licences you might need for business activities. Also check out the government's official page on "Driving licences for taxis and private hire vehicles" for information on Private Hire Vehicle (PHV) licenses.
As an aside, a perspective on how times have changed:
- Don't get in strangers' cars
- Don't meet people from the Internet
- Literally summon strangers from Internet to get in their car