This is a great question. Answering the question "how well could different types of people read" is a challenging one historians have spent a lot of time on. As you can probably imagine it isn't always possible to use primary sources to come up with statistical data that meets modern standards, this really doesn't matter though because as we will see the whole concept of 'literacy' changes pretty drastically over this period. "Europe post-1500" is by far the most well documented and researched category of history there is (legacies of colonialism) so we can definitely arrive at answers to our question by taking a brief survey of the scholarship.

Let's start with a recent article surveying midcentury English literacy. The author's introduction familiarizes us with some of the differences between our definition literacy and how it is applied in this period. First the Victorians saw reading and writing as two distinct skills, to be taught sequentially. Some authorities, like the Methodist Church (after internal debate,) specifically excluded working class people from learning to write; the reasoning being that while reading scripture has a religious justification writing was a "worldly" economic skill that would corrupt and shouldn't be taught by the Church. Until the introduction of mass state education very late in the period (1870) people unable to pay for education receive it primarily from churches that offer Sunday school, day school, or charity school services. Even if they are taught to write chances are their attendance leaves a lot to be desired to say nothing of the quality of instruction.

Then again a "respectable" working class family would take pride in its strict observance of Sunday services, their children would be relatively educationally advantaged. Another factor promoting literacy is the explosion of newsprint, which makes reading material far cheaper and more accessible than it had ever been before-- we know that even illiterate people were big consumers of print culture, even in the poorest slums every pub will have a newspaper and someone willing to read it to you for a pint.

What all that creates is a large population of "semi-literates" who have significant but highly uneven literacy skills. Quite commonly someone who can read but not write, or someone whose literacy skills are limited to specific familiar materials (Bible, craft manuals)

The article I linked has some numbers for English literacy buried within all the caveats, I'll excerpt them but adding my own that they really need to be understood in the context of her paper. Between 1844-49 the literacy rates for unskilled labourers tended to be between 30-40% and skilled labour between 60-70%.

Now let's compare Ireland to England. Literacy rates were relatively high in Ireland; around 47% in 1841.. Though Ireland is certainly languishing through the entire first half of the 19th century it has literacy rates that are comparable to England's, which is even more surprising given the ongoing Anglicisation transition from Gaelic to English. Honestly Irish history is one of my weakest areas (it's a big empire!) and I won't venture an explanation for why Ireland continued to do so well in literacy through this period. I know we have some very talented Ireland people on the board, hopefully my answer can stand as a background for their explanation.

By 1800 there were no legal barriers to the education of Catholics, in fact the British administration favoured education as they believed it would promote social stability. There is quite a lot of evidence from the 1841 census and the 1825-6 Commission on Irish Education which surveyed every parish in the country.

Education depended on social class and religion. There were a wide variety of schools for Catholics and Protestants educating about 500,000 children in 1825. Poor children, those whose fathers were labourers or cottiers (subsistence farmers) would have gone to a pay school (also called hedge schools). This would have been run by a local school master who was paid by the lesson and in the form of food or goods. The school would have taught reading,writing and arithmetic but the standard was very variable. Some schoolmasters were superb, others barely had primary education themselves. Books were brought by the children and were anything from newspapers to catechism.

Remember that when we talk about literacy we mean reading and writing English. Many of the poor would have spoken Irish as a native language but parents, experts, Church leaders and politicians were unanimous that they learn English to improve their prospects. This meant that they were learning reading and writing in a second language which may have hindered some.

There are several reasons why a poor child might remain illiterate. Their family might not have been able to afford even the small payments for school, the local school master could have been bad and they might have only attended sporadically as they would have worked on the land from very young. A guesstimate by Niall Ó Cíosáin is that about one in six labourers could read.

The biggest obstacle to mass education was rivalry between the churches. The government established a free primary education system in 1831 (far earlier than in England) but all the Churches objected to the fact that it was non-denominational. It took decades for the state to give in and set up a church dominated system.

Sources:

Garret FitzGerald, ed. Irish Primary Education in the Early Nineteenth Century: An analysis of the first and second reports of the Commissioners of Irish Education Inquiry, 1825–6

James H. Murphy (ed), The Oxford History of the Irish Book, Volume IV: The Irish Book in English, 1800-1891

W.E. Vaughan (ed),The new history of Ireland vol. V Ireland under the Union 1801-70