In multimember constituencies of the House of Commons, how many votes did each voter have?

I am not going to try and answer for the earlier periods (at least not just now - writing this from the train...) Prior to the Reform Acts, properly contested elections were of course less common.

By the nineteenth century, the general system was that each voter had two votes. They could vote for any two of the candidates. A voter could also "plump", vote for only one candidate (counting as a single vote).

There was not an official 'slate', but by the post-Reform period a contested election would usually be contested by two members of each party, and those affiliations were known. The two candidates with the most votes won seats. Split returns - say one Conservative, one Liberal - were thus possible.

Interestingly, as the secret ballot didn't appear until later in the century, quite a few poll books survive - you can thus examine how many people voted a split ballot versus a straight one. I'll edit this when I can find a link.

**edit ** right, some pre-Reform c19th examples with poll books

Westmoreland, 1818 general election

Note that this is for Westmoreland voters in Appleby, not for the Appleby seat. It had three candidates, two incumbent brothers who were locally poweful Tories, and a Whig challenger. This was the first properly contested election for the seat in living memory. The challenger took about 27% of the votes cast, not a bad total, but -

It was a straight contest between Brougham’s ‘Blues’ and the Lowthers’ ‘Yellows’ and Thanet, for one, thought it disastrous that Brougham was unable to obtain more split votes—he received only 56 of them at the poll. (HoP)

In other words, the vast majority of votes went either for the challenger or for the incumbent(s).

Aylesbury, 1831 general election

Aylesbury had three candidates, two votes. Nugent and Rickford were both incumbent reformists, opposed by a single candidate, who "professed to approve the principle but to object to many of the details of the reform bill, and declared his support for the constitution in church and state and for the agricultural interest" - so a bit more Tory-leaning. The voting patterns are described at length by the History of Parliament:

[Rickford] was supported by 86 per cent of those who voted, Nugent by 53 per cent and Kirkwall by 44 per cent. Only 199 (18 per cent) of the voters cast plumpers (43 for Rickford, 58 for Nugent and 98 for Kirkwall, who derived 19 per cent of his support from this source). Of the 951 who gave split votes, 540 (47 per cent of those who voted) voted for Rickford and Nugent (55 per cent and 89 per cent of their respective totals) and 403 (35 per cent) divided for Rickford and Kirkwall (41 per cent and 79 per cent of their respective totals). Six-hundred-and-forty-one (56 per cent) voted for one or both of the reform candidates, while only Rickford's plumpers (nine per cent) voted unequivocally against reform. Of the 314 Aylesbury residents who polled (28 per cent of the total), 93 per cent voted for Rickford, 57 for Nugent and 42 for Kirkwall; while the freeholders gave their support in the proportion of 83, 51 and 45 per cent respectively. Thus Rickford was significantly stronger in the borough than with the voters as a whole, but there was no marked bias in favour of Nugent or Kirkwall in either quarter.

So, unlike Westmoreland, there were very few purely "challenger" votes, with many of his supporters choosing to split their ballots and vote for one of the incumbents as well. A very different dynamic.

Hull, 1832 & 1835 general elections, 1835 by-election

In Hull, the poll books are a bit more confusing. They only list a single name (well, initial) against each person. But each elector seemed to have two votes - there were 3730 cast in 1826 against only 2298 electors - so I'm not sure quite what's being done here.