I tend to avoid a big meal before the theatre, so I either get something light before the show or choose to go post-theatre.
Some of the restaurants that I’ve enjoyed solo before or after the theatre:
1) The Palomar in Leicester Square - Really interesting Middle Eastern small plates, sit at the bar and watch the cooks
2) The Barbary - owned by the same group that runs the Palomar, similar theme
3) The Opera Tavern - Spanish and Italian small plates, very nice selection of wines by the glass
4) Masala Zone - It is a chain, but I enjoy it.
5) Dishoom - Trendy Indian food. The Covent Garden location nearly always has a queue so I generally avoid that location. I actually prefer Dishoom for lunch as it tends to be less crowded.
6) Nopi - I’ve eaten there a number of times. Be aware that the pre-theatre menu they advertise is not available to solo diners. I found that out when I went once before a show and they told me it was only available for ”sharing”.
Some places that others have recommended but were just ok for me:
1) The Delaunay - Austrian food, beautiful setting, but I found it very noisy and I got the schnitzel which was nowhere near as good as the schnitzel I had in Austria
2) Brasserie Zedel - French bistro food, good value. When I was there last October post-theatre, the service was just terrible. Maybe I was just there on a bad night as a number of people have recommended this place in the past.
As far as London vs New York, I prefer London. To be fair, I’ve only seen 4 shows in NY and I have seen more than 50 shows in London over the last 25 years. I think the quality of the performances is comparable. I give London the edge mainly because it is such a comfortable place for a solo traveler to walk around post-performance; I always feel very safe after dark. Plus, they sell ice cream at the interval! The venues in London vary in quality. Some are big and luxurious (e.g., Theatre Royal Drury Lane). Some are rather run down and in need of renovation. Some have poor ventilation and can be uncomfortable when it is hot. Some seats have very little legroom or poor rakes. I do rely on theatremonkey for seating recommendations.
I hope you enjoy your stay!