Rain in London isn't an event, it's a baseline. Pretending otherwise is how you end up sheltering under the awning of a Pret in Holborn explaining that this wasn't the plan. Build the rain in from the start and the date gets easier, not harder.
Pick a museum with a bar
The trick is choosing a museum you can actually drink in afterwards. The Wallace Collection on Manchester Square has a glass-roofed courtyard restaurant that feels like a greenhouse in a thunderstorm, and the museum itself is free and small enough to do in an hour. Sir John Soane's Museum on Lincoln's Inn Fields is even better in bad weather, all candlelit corners and crammed antiquities; book the evening tour if you can, then walk three minutes to the Seven Stars on Carey Street, which is the kind of pub that makes rain feel like part of the atmosphere.
The Tate Modern works for the same reason. The Members Room bar on the sixth floor has a view of St Paul's through the rain that's better than the view in sunshine.
Do something with your hands
A weather-proof date that involves doing rather than watching always lands better. Drink, Shop and Do near King's Cross runs craft evenings most nights, from clay modelling to lino cutting, for around twenty pounds a head. London Terrariums in Bermondsey runs two-hour workshops where you build a small ecosystem in a jar, which is a more interesting souvenir than a fridge magnet.
For something less crafty, Swingers in the City or in West End does indoor crazy golf with proper food and drink. It is mildly ridiculous, which is the point on a date that needs to lighten up.
Cinema, but specific
The Prince Charles off Leicester Square shows everything from new releases to themed weekenders for under ten pounds. The Castle Cinema in Homerton has sofas and table service. The Electric in Notting Hill has armchairs and footstools and a bar at the back; it is overpriced and worth it once. The Curzon Bloomsbury has the best programming in central London and a basement bar that is empty enough at six on a weeknight to actually talk.
Avoid the big chains in Leicester Square on a date. The seats are fine; the surrounding experience is not.
A long lunch that becomes the afternoon
Rain is permission to eat slowly. Quo Vadis on Dean Street does a set lunch that runs to two thirty and the room is warm and lined with stained glass. St John on St John Street does the same job in a whitewashed dining room that's improved by grey light through the windows. Both let you sit with a bottle of wine for three hours without anyone hurrying you out.
If you want cheaper, Koya on Frith Street does udon in a steamed-up little room that is the best place to be in London during a downpour for under twenty pounds.
Bookshops that absorb a whole afternoon
London Review Bookshop on Bury Place has a cake shop attached and chairs you can actually sit in. Daunt on Marylebone High Street has the long oak galleries that everyone photographs; go on a weekday and they're empty. Foyles on Charing Cross Road has five floors and a cafe at the top, which is enough material for two hours of browsing and a coffee.
The move is to give yourselves a task. Each pick a book for the other under a tenner, meet at the till. It is a better get-to-know-you than any list of questions.
The pub, properly
When all else fails, commit to a single pub for the whole evening. The Lamb on Lamb's Conduit Street, the Harp off Charing Cross, the Holly Bush in Hampstead, the French House in Soho. Pick one with no music and a fire if possible. Order food. Stay for hours. The reason this works in the rain is that leaving feels like a decision, and decisions create momentum.
What to wear, briefly
London rain is rarely heavy and almost never warrants an umbrella that won't fit on the tube. A coat with a hood beats an umbrella every time. Suede shoes are a mistake from October to April. None of this is romantic advice, but the date where one of you is visibly soaked and miserable is a date that ends early.