What are the quickest/most prolific flowers to grow?

Hardy and half hardy annuals.

Larkspur (also good for drying as well)
Cornflowers
Nigella/ love in the mist (good for seed heads as well)
Cerinthe (filler type)
Scabious (some are good for seed heads like ping pong)
Calendula / pot marigold
Ammi Majus & Ammi Visnaga (filler type) you could also forage for cow parsley
Gypsohilia (although I don't do very well growing it)
Clary sage
Euphorbia (but its sap is an irritant I personally wouldn't grow/sell it)
Anthirrinum/snap dragons (look for tall varieties)
cleome (they have spikes)
cup and saucer plants
cosmos
zinnia
tithonia
rudbeckia
sunflowers (esp. small headed ones like vanilla ice)
Spanish flag
statice (can be used dried)
helichrysum/ straw flower (can be used dried)
nicotiana
sweet peas (price will depend on quality and length of stem - there is a bit of an art to growing 'good' sweet peas) If you want to get into growing tips look for Roger Parsons sweet pea book.

make sure the flowers are conditioned for several hours before selling. Picking first thing in the morning and last thing at night.

good flower seed sites (IMO)

chilterns - Veg Seeds | Flower Seeds | Plant Seeds | Hanging Baskets | Vegetable Seeds | Buy Flowering Plants from Chiltern Seeds
higgledy piggledy - The Higgledy Garden Shop | Higgledy Garden
sarah raven (be warned they tend to be on the expensive side, she also does some fantastic vids for arranging and bits and pieces) - https://www.sarahraven.com/

british florists and flower growers that are worth a look (IMO)

Common Flower Farm (Georgie Newbury she also has a book that is worth checking out at your library about starting a flower farm with lots of practical tips) - Common Farm Flowers Common Farm
Green and Gorgeous - Flowers - Green and Gorgeous Flowers
great British florist (they have some you tube vids that might be worth checking) - Flower Delivery for British Flowers | Great British Florist
Buckets of blooms (newish company) - http://www.bucketsofblooms.co.uk/
Garden gate flowers - story
Floret (American but lovely) - https://www.floretflowers.com/

Try and stop there with the sites if you can. Most tend to be wedding focused and start to get a bit samey.

If your friend carries on then look at sowing bi-annuals june - july and some of the hardy annuals in September to get an earlier crop. Also chrysanths and dahlias will extend the season but do require a bit more effort on the yearly propagating side.

Sorry for the lengthy post but I have been looking into this for a few weeks now.

Oh my goodness that is an INCREDIBLY helpful and generous post - thank you very very much! I'll pass all of that on - thank you!

Yes, shock horror I forgot something. Herbs can be a great addition to flowers depending on how and where your friend sells them. Dill, fennel, mint, lemon balm etc.

You missed a few- Lavatera, Escholtzia,Love Lies Bleeding,Clarkia.

^^^^and stocks, how can I forget stocks. Runs off and bangs head on wall

Cosmos gets my vote for producing a generous quantity of flowers

Stupid question time is lavatera the same as mallope?

NB there is a perennial/tree shrub lavatera as well.

A vote for Cosmos here

Tree Mallow, just chopped 4, spring up everywhere in my garden.

A few more for you

Bells of Ireland (filler/flower)
Candytuft (filler/flower)
Salpiglossis (an annual alternative to Peruvian lily/ Alstroemeria)

On a side note there is legislation on the use of manure. Unfortunately Defra sucks and the legislation is not very clear and certainly doesn't try to help the little people. I don't think many if any actually adhere to it but it does exist. Basically only 10% of a crop (inc. flowers) can be grown on manured soil but it doesn't give quantities or period of time. Lots of long winded bits about soil management and records and stores should be temporary and away from water course. I haven't checked lately but I would guess chicken poo is not permitted for some time with the bird flu issues. I am just going with green manures to cover my bum.