This is the earliest street directory I have found so far which also includes publican detail.
London Street Listings in 1832.
Haberdashers place, Hoxton
Haberdashers street, Hoxton
Haberdashers walk, Hoxton
Haddon place, see Great Waterloo street
Hadlow street, Burton crescent
Hagley place, see New Kent road
Hales terrace, see Commercial road, Limehouse
Half Moon passage, see Leadenhall market
Half Moon street, Bishopsgate street
Half Moon street, Piccadilly
Half Nichols street, Bethnal green
Hall place, see Kennington lane
Halken street, Belgrave square
Ham yard, Great Windmill street
Hamilton place, Kings cross, New road
Hamilton row, see Battle Bridge
Hammet street, Minories
Hammonds court, Mincing lane
Hampton street, Walworth
Hand court, Holborn
Hand court, Upper Thames street
Hanover place, see Old Kent road
Hanover place, see Regents Park
Hanover square, Cripplegate
Hanover street, Hanover square
Hanover street, Long acre
Hanover street, Walworth
Hans place, Chelsea
Hans street, Chelsea
Hanway street, Oxford street
Harcourt buildings, Temple
Harcourt street, New road, Marylebone
Hardwick street, New River head, Spa fields
Hare court, Aldersgate street
Hare court, Temple
Hare street, Bethnal green
Harley street, Cavendish square
Harp alley, Shoe lane
Harp lane, Great Tower street
Harpur street, New Kent road
Harpur street, Red Lion square
Harrison street, Grays Inn road
Harrow road, Edgware road
Hart street, Bloomsbury
Hart street, Covent garden
Hart street, Cripplegate
Hart street, Crutched friars
Hart street, Grosvenor square
Hatfield place, see Westminster road
Hatfield street, Golden lane
Hatfield street, Stamford street
Hatton court, Threadneedle street
Hatton wall, Hatton garden
Haydon square, Minories
Haydon street, Minories
Hayes Court, Soho
Haywards buildings, see City road
Heartley place, see Old Kent road
Heath street, Commercial road
Heathcote street, Mecklenburg square
Heddon court, Regent street
Heddon street, Regent street
Hedley place, Hoxton fields
Helmet row, St Lukes
Hemmings row, St Martins lane
Heneage street, Whitechapel
Henrietta street, Brunswick square
Henrietta street, Cavendish square
Henrietta street, Covent garden
Henrietta street, Manchester square
Henry street, Grays inn lane
Henry street, Hampstead road
Henry street, Pentonville
Henry street, Waterloo road
Henry street, St Lukes
Hercules buildings, Lambeth
Hercules court, Thredneedle street
Hereford street, New road
Hermes street, Pentonville
Hermitage street, Wapping
Little Hermitage street, Wapping
Hertford street, Fitzroy square
Hertford street, Mayfair
High street, Hoxton
High row, Knightsbridge
High Timber street, Upper Thames street
Hill street, Berkeley square
Hill street, Finsbury
Hill street, Great Surrey street
Hill street, Walworth road
Hind court, Fleet street
Hinde street, Manchester square
Hindon street, Pimlico
Hingage street, Bevis Marks
Holborn, High - or High Holborn
Holland street, Great Surrey street
Hollen street, Soho
Holles street, Cavendish street
Holles street, Clare market
Holloway street, Commercial road
Holybush gardens, Bethnal green road
Holybush place, Bethnal green road
Holywell lane, Shoreditch
Holywell mount, Shoreditch
Holywell row, Shoreditch
Holywell street, Strand
Holywell street, Westminster
Homer place, see New road, Marylebone
Homer street, Marylebone
Honduras street, St Lukes
Honduras terrace, see Commercial road
Honey lane, Cheapside
Honey lane market, Cheapside
Hooper square, Goodmans fields
Hooper street, Clerkenwell
Hop gardens, St Martins lane
Horseferry road, Limehouse
Horseferry road, Westminster
Horsemonger lane, Borough
Horse Shoe alley, Wilson street, Finsbury
Horse Shoe yard, Lower Brook street
Hosier lane, West Smithfield
Horsley street, Walworth common
Houghton street, Clare market
Howard street, Strand
Howford buildings, Fenchurch street
Howland mews West, Middlesex Hospital
Howland street, Middlesex Hospital
Hoxton market, Hoxton
Hudson terrace, Westminster
Huggin lane, Wood street
Hull street, St Lukes
Hungerford market, Strand
Hungerford street, Strand
Hunter street, Brunswick square
Hunter street, Old Kent road
Huntingdon street, Kingsland road
Huntley street, Bedford square
Hunts court, Castle street, Leicester square
Hyde place, Hoxton
Hyde street, Bloomsbury
Hylords court, Crutched friars