There probably were always professional soldiers in post-Roman Britain, but the scale of these armies (both in raw numbers and in territorial reach) would have been severely restricted to the 'small worlds' of Britain's river valleys and many small kingdoms for many centuries after the Roman withdrawal.

By the fifth century, many wealthy Roman elites had begun to keep small groups of professional bodyguards / retainers in their households, called 'bucellarii' ('biscuit eaters'). Powerful elites, like Justinian's general Belisarius in the early sixth century, employed enough bucellarii to act as a professional military force. Small elites, like those who remained in late Roman Britain after the official legions left in the late fourth or early fifth century, probably employed far fewer retainers. But these private soldiers almost certainly formed the power nuclei around which local elites maintained some semblance of political order in Britain through to the end of the fifth century.

By the end of the fifth century, there's archaeological evidence which suggests that responsibility for fighting was starting to be spread across a broader cross-section of society: half of the men buried after c.480 had spears in their graves, which probably relates to their status as part-time 'warriors' (though many of them likely never fought, and their primary concern would have been farming grain and cattle). This diffused responsibility for military service is probably the earliest form of the 'fyrd' (community militia) that Anglo-Saxon England is famous for depending upon in later centuries. But even then, there's evidence that elites continued to support professional (or at least, priviledged) soldiers: the first 'Anglo-Saxon' law code, written in Kent, mentions 'loaf eaters' as a special social class connected with the king (the english word 'lord' is etymologically derrived from 'hlaf', loaf: the lord is the man who feeds the bucellarii). These private 'armies' were small, local, and personal, but so were politics in post-Roman Britain. An 8th century English law code defines an army as any group of armed men larger than 35; warfare was small scale in post-Roman Britain, and a lord with a few dozen well-fed retainers might have hoped to carve a valley out for himself. And a few of these local warlords won big and founded the kingdoms that were, centuries later, united into Anglo-Saxon England.

But if you want a full national army, you'll have to look much later: England wouldn't be united under a single king until the tenth century.

I would recommend Gerrard, The ruin of Roman Britain, if you want to read more on this. Halsall, Warfare and society describes more generally the role of private semi-professional armies in the political landscape of the post-Roman west.