As an answer specifically for Britain, native horses seem to have become extinct for a few thousand years before being reintroduced, presumably by being transported here by boat from Europe ^edit: ^see ^below. As temperatures began to rise after the end of the Ice Age and into the Mesolithic there's evidence for horses being hunted for meat at occupation sites, but no evidence of them being used for transport either to ride or as pack animals. They don't reappear again until around the Bronze Age, and are thought to have been domesticated by 2000 BC. Horse remains become visible in the archaeological record again around that sort of time, sometimes incorporated into earthwork monuments like Barrow C at Snailwell in Cambridgeshire. There also seems to have been some association with horse use and the Beaker using continental culture that arrived in Britain c.2500 BC.

Harness fittings for horses are visible in the archaeological record in Britain from c.1000 BC suggesting they were being used for transport by at least this point, and other objects like cheek pieces and phalera (small decorative discs associated with riding) begin to be included in metalwork hoards. Archaeologically it's difficult to differentiate between pack animal use and riding, especially given that any early horse tack would likely have been made from cordage, leather, and/or wood which doesn't preserve as readily as metal. It's quite possible that they were being used for transport before the associated metalwork becomes archaeological visible.

Around this time some kind of wheeled transport, likely wagons or small carts appear. The earliest preserved wheels in Britain date to c.1250-800 BC, but the exact use of these carts isn't something we have a good understanding of. Returning to the idea of continental cultures taking up horse use first, the earliest wheels in northwest Europe date from c.3000 BC suggesting Britain was comparatively late to the idea of the use of horses in conjunction with wheeled transportation.

Sources:

Darvill, T., 2010, Prehistoric Britain, 2nd edition, London: Routledge.

Milner, N., et al, 2013, Star Carr: Life in Britain after the Ice Age, York: Council for British Archaeology

Edit: Apologies, I've misinterpreted your question. I thought you were asking about horses as transportation, not transporting horses. The 'how' of how horses were reintroduced to prehistoric Britain isn't something I know a lot about, but I can give it a go based on what I've outlined above. It is possible that horses didn't become completely extinct, but to my knowledge based on the sources given above there's no archaeological record of their presence between the Mesolithic and the Bronze Age.

For context, Britain became an island between c.7000-6000 BC, with parts of what we now call Doggerland persisting up until c.5000 BC at the latest. Doggerland is the name given to the stretch of land that connected Britain to the European mainland, it would have made transport from the Rhine area through into the south of England fairly straightforward as well as providing a general route from Britain to Europe, and based on some of the finds dredged up accidentally by trawlers it seems to have been inhabited.

After Britain became an island, if we accept that horses were reintroduced by boat rather than surviving in a way that hasn't been archaeologically preserved, it would have been fairly difficult. Horses at the time were small and stocky compared to what we think of when we think of horses now, more akin to working animals than what you might see doing dressage at the Olympics. Given the size then of a foal, it may have been feasible over a period of time once a breeding population had been established.

Boat technology of the period seems to have been more more advanced than one might have thought, with plank-built boat remains from c.1870 BC being found in Yorkshire. There are some examples that date a little earlier, but for plank boats that's what you're looking at for Britain. By c.1575-1520 BC we see a plank-built boat named the Dover Boat over 2m wide and at least 14m long which may have been able to cross the channel. Dugout canoes are more common and can be found with earlier dates, but it's unlikely they would have been able to manage a successful cross channel journey, let alone carrying a significant number of horses given how dangerous the channel can be.