SinghKing
Banned
Well, first, as noted by SinghKing, Liverpool surpassed Bristol before 1800.
Second, it wasn't just Liverpool, but also Manchester and Birmingham. If you want Bristol to be the second city, you need to explain why it's bigger than Manchester and Birmingham, too. 19c Liverpool had more people in the city proper than Manchester and Birmingham, but its metro area was and still is smaller.
Northern England's industrial dominance comes out of geographic factors: it is drier than the rest of UK, which makes it less agriculturally productive, so in the early stages of the Industrial Revolution, it specialized in industry (Lancashire) and mining (Yorkshire) instead. The same is true of the West Midlands: Birmingham was located next to iron deposits, which led to early metalwork industries like gun manufacturing, eventually making it a steam engine powerhouse.
So if you want Bristol to remain the second city, you need to somehow shut down both centers of British industrialization.
Now, this is perfectly plausible! Figure out a POD that delays British industrialization. Perhaps the Industrial Revolution started in the Netherlands, which had a 150-year head start on Britain even in OTL. Lancashire remains a backwater, and nobody needs a city on a watershed between three counties. In this setting, it's not hard for Bristol to overtake York and Norwich as it did in OTL - Britain is still going to have a maritime empire, it's just going to be more like the declining Spanish empire.
So, going with this, let's say an unspecified late-17c POD made Britain incapable of wresting control of maritime trade from the Netherlands, and conversely prevented Dutch stagnation. In OTL, even with 18c Dutch stagnation, the Netherlands' GDP per capita income remained higher than Britain's throughout the Industrial Revolution. So if industrialization began in the Low Countries, it happened earlier - let's say 50 years in advance of OTL, to give metalwork time to catch up to allow steam engines. Let's also say Belgium industrializes as early as the Netherlands - either the early-industrializing Netherlands is capable of conquering it (or maybe of splitting it with France along linguistic lines), or Liege expands its industry to compete with the Netherlands.
Around 1750, the *Netherlands has a growing urban economy, with so much population growth that it dumps settlers all over the world, including New Netherlands, which in the 1770s gets to leapfrog British North America to the Midwest using the Erie Canal. In 1780, railroads are developed, starting from Walloon mine-to-port lines and expanding to replace Dutch canals since they're faster and don't freeze in the winter.
By 1800, the less developed monarchies of Britain and France want a piece of the Dutch industrial economy, but aren't really capable of conquering any of it. They hire Dutch engineers to build them their own factories and railroads and steamships, same way OTL's late-20c developing countries hire first-world firms to build factories and architectural megaprojects. Britain is interested in connecting London with the three biggest regional cities: York, Norwich, and Bristol. The London-York line flops because of the locality of travel (read Andrew Odlyzko for much more detail - it's all online). The other two lines succeed, and Britain slowly develops a specialized maritime economy around Bristol, and undergoes urbanization and income growth, although it lags behind the Low Countries. Just as OTL's France didn't have a big change in city size ranking, except specifically that the Occitan-speaking cities went into relative decline in the Vergonha, TTL's Britain has roughly the same city size ranking as in the 18th century.
Well, that's a fair suggestion. Still, I'd prefer to do this without having to pursue a Brit-screw solution- hoping to err toward the other side, and to boost the British economy and population size ITTL compared to IOTL. I've been wondering though, about the importance of the role which could be potentially be played by Edward Parry's discovery of what would IOTL be re-discovered and patented in the USA as Fiberglas (perhaps patented ITTL as Glasswool or Cottonglass?), 99 years earlier ITTL.
It's pretty important to remember which era we were in at the time- the Little Ice Age was still ongoing, and it would take another 20 years or so until temperatures started to rise again. In London, in and around this era, between 1849 and 1860, Victorian newspapers reported that cold snaps were responsible for increasing the average mortality rate by around 25% every annum- as a conservative estimate, which didn't take the unregistered street population into account. And what is Fiberglas most commonly used for IOTL? Housing insulation. If they pioneer the use of the material to insulate housing as the Little Ice Age takes its final and most intense dip in temperature, going into the early 1850s, then South Wales and Bristol will have the warmest homes in the world at the time when it matters most. Just through the benefits of the improved housing insulation alone, over the space of only ten years, they could plausibly reduce mortality rates in the city and its immediate environs to levels which would only be seen at the dawn of the 20th century IOTL. And that could have a huge impact on the population growth of the city.
Bristol was the primary port and transport hub for the West Country, which was an extremely popular tourist destination for people from London and the Midlands once the transport links were in place (becoming known as the 'English Riviera'). Several Victorian resort towns sprang up in this region, and large numbers of holiday homes were constructed in the region, with these wealthy tourists swelling its population every summer, before returning home in the winter. But ITTL, as the first region to feel the benefits of modern housing insulation, the houses in and around Bristol would be the warmest and most comfortable residences in the entirety of Great Britain- during the Little Ice Age, when cold mortality rates were higher than they would ever be again.
Why would those tourists want to return home in the winter, running the very real risk of catching their death of cold every year? ITTL, if they relocate to this region, and take permanent residence there, they can have the best of both worlds- relaxing in their seaside resorts in summer, and living in warmth and comfort unparalleled anywhere else in the United Kingdom (until the innovation of house insulation spreads further afield- by which time, the Little Ice Age will have largely abated anyway) during the harsh winters. And unlike the French Riviera, they'd still be residing in the British Isles, enabling them to continue running their British business affairs from here in the era prior to widespread telecommunication.