I'm currently reading Longbourne which is set in the world of Pride and Prejudice and the central character, a servant of about 18, is depicted as being able to read. It struck me as sounding wrong, because I can't imagine that people like that ever went to school, but the rest of the book is well-researched so I'm not sure.
More likely than not; the literacy rate in England in 1800 was between 50-60%.
What type of servant? Ladies' maids would have been required to read aloud as part of their duties, as would quasi-servants like governesses and companions. Housekeepers, butlers, and cooks (and any lower servants aspiring to hold these positions) would have been literate - they each dealt with recipes, market lists, receipts, etc. We'll have to count in dairy maids too, as they also kept lists and ledgers.
I wrote a longer summary about the state of education in the UK ca. 1800 here. There are a few ways in which a working class person might have learned to read: from family members, at a dame school, through the church, or at a charity school (the most modern-school-like, and the least likely). Servants may also have learned on the job - especially if they entered service as an apprentice at a young age. Depending on the type of servant, there's a fair amount of down time in the afternoons and early evenings when this learning might have taken place.
So yes, it's totally plausible that even the lowest of housemaids would have at least rudimentary literacy.