It seems unlikely...
With the exception of Henry Taylor's suggestion (which is intriguing to say the least)
The Roman empire collapse due to many contributing reasons, and the British empire rose to power due to many contributing reasons, The Fall of Rome, which was sacked by many different groups, including the Angles, Saxons and Franks, was a important event, without it, the Angles and Saxons couldn't have settled and grown in the British isles, and the Franks couldn't have settled and grown in France, without the Franks spreading and splitting (creating many little kingdoms and nations such as the Normans) then the Norman invasion of Britain would likely never have happened, without it, there would be no Britain, no Britain no British Empire.
So realistically the Western Roman Empire really had to have fallen for the British Empire to have even existed, but the Eastern Roman empire could have survived the Fall of the Rome, and it did, ruled from Constantinople (founded as Byzantium) this became the Byzantine Empire (although still frequently referred to as the Roman Empire), this empire really wasn't the same as the classic Roman empire you probably want, as they were Orthodox Christianity and spoke Greek not Latin. I'll chuck the wiki page in rather than recount all the factors of this
But if the Byzantine Empire is good enough for your story then you need to stop the Ottoman Turks from conquering them, but they only managed this due to being weakened by a series of crusades between 1057 and 1204 performed by several nations including importantly England... part of the British Empires important early power comes from doing irreparable damage to the Eastern Roman Empire...
This is why i thinks its unlikely, and as an aside, Empires tax goods and services even if they are just passing through, the British Empire wouldn't have accepted paying extra to the Roman Empire. they already had enough issues with other countries hence the amount of Wars they started (I'm British myself, don't argue that we didn't start them, yes some were started by others but we started a lot of them) they wouldn't tolerate yet another competitor to their power
But as i said above Henry Taylor's answer is a real theory, a crazy one, but bizarrely its a crazy and at the same time plausible one, so i guess it depends on how you feel about it
Just as an alternate suggestion
In the Fallout computer games, the world has suffered extreme nuclear fallout for a long time, and there are a few different factions trying to restore order and even more trying to take power. one of the factions that (from there own warped perspective) that are trying to restore order is Ceasar's Legion, the entire faction grew out of a historian, Edward Sallow, who was captured by the Blackfoot Tribe, while he was being held for ransom, he ended up explaining the roman empire to the tribe, along with other information like gun maintenance and small unit tactics, they made him their leader and he took the name Caesar, and organised the tribe into a legion (hence the name)
So my question is do you need the Roman Empire to survive at all, or can someone pick up some history books and realize that when it came to Pre-Firearm fighting... the Romans really were very advanced. if your post apocalypse tribes either had no access to firearms or it has been so long there are no bullets left, then it is actually believable that an organised group would probably make roman weapons and use roman tactics, and it wouldn't take that big of a leap for them to take the name and adopt the culture of their greatly successful icons