London has a constant flow of business travellers who are on their own and likely to be seen dining single. Also, there are several neighbourhoods that cater to the singles crowd. As a consequence, there is no stigma attached to eating alone like there might be in other cosmopolitan cities. But more to the point, London restaurants will happily accommodate anyone with the capacity to pay and is reasonably dressed. For starters, you can read Time Out's article on London's best restaurants for dining alone. This article presents a few upmarket restaurants that will seat singles at their bar.
Caveat: London restaurants come and go, so there's no guarantee that a list of specific restaurants will be helpful three months from now.
So rather than giving out links to specific restaurants, I will list some neighbourhoods and locales where there lots of restaurants to choose from and explain why they (in particular) are suitable for single travellers.
Marylebone High Street This neighbourhood is brimming over with great restaurants that are small and comfortable. There are lots of single pensioners in the residential flats at the top of the street, but more interestingly it runs parallel to the medical district on Harley Street and thus is a favourite haunt for doctors and nurses to eat lunch or dinner.
St Christopher's Place This is a small area abutting the south end of Marylebone High Street adjacent to Oxford Street (the shopping district). It is basically a pedestrian area filled with restaurants that cater to shoppers wanting to eat lunch or dinner. It's fine to dine either outside or in as a single, but towards about 9 PM or so, it becomes part of the 'singles scene' and is more likely to be populated by groups of 5 - 8 people.
Pavilion Road in Knightsbridge also caters to professionals and shoppers who are looking for an up market restaurant. As a single, it's best to hit this area early because after about 9 or 10 PM it becomes populated by the 'Sloan Rangers' (the posh singles set like Prince Harry when he was younger).
And of course the London Docklands, which is further away from the centre, but is often frequented by bankers and other professionals who worked late and are looking for an appropriate place for dinner before heading home.
That's not an exhaustive list, there's also the western end of King's Road in Chelsea and Brick Lane in 'the city'. Other answers have mentioned Soho. These are neighbourhoods where you can find excellent restaurants where a single diner can get a nice table and not appear out of place at all. And if you find yourself overwhelmed, you can always take a seat at the bar.
Finally, you can always check Trip Advisor for any specific restaurant and see what others have experienced. It's not perfect but might save you from a "let down".