How are housing costs reflected in factor shares of GDP? The labour share of GDP in the UK is around 55% (Source).
Presumably all labour is supplied by households and all households need a house.
The average percentage of total monthly household income spent on private rent is around 30% (Source). For the sake of simplicity I assume that this percentage is the same for the imputed rent of owner-occupiers.
How do housing costs enter into the factor shares of GDP?
If a household earns 30k/year from labour and pays 10k/year rent, does that result in 30k x 0.7 = 21k towards labour share of GDP and 30k x 0.3 = 9k towards land/capital share of GDP?
They enter through the output. By definition the labor share of GDP is: $w/Y$ where $w$ is factor income that is by statisticians assigned to the labor. Note  this does not necessarily correspond to the economic concept of labor as the statistical labor share of GDP calculation lumps together multiple factors of production and calls them labor - the same way as ‘capital’ share lumps together returns to land, capital, entrepreneurial labor etc.
Now the labor share of income is derived from incomes, but at the macroeconomic aggregate level all spending must be equal all income (although statistically there might be discrepancies due to measurement error). So the numerator can be replaced by total spending in the economy giving us:
$$\frac{w}{Y}=\frac{w}{C+I+G+NX}$$
Where $C$ is consumption spending, $I$ is investment spending, $G$ is government spending and $NX$ is spending on net exports. Now expenditure on rent is included in the $C$ part so that is how these expenditures enter the labor share of income calculation.
However, the back of an envelope calculation at the bottom of your question is not correct for several reasons.
First, If the labor income is 30k then that 30k is already the result of multiplying GDP by labor share i.e. it is $\frac{w}{Y }(Y)=w=30k$
Second if the labor income per year is 30k and rent is 10k and we assume that there is nothing else going on in the economy,  then it is the 10k that goes to the capital income and  calculation of its share. To be more precise here the total output by income approach would be 40k (30k wages 10k rent - this would by the way be also equal to all spending where rent would be part of $C$ and the other income also part of $C$ or other components of aggregate spending) so the labor share would be 30k/40k or $75\%$ while the capital share would be 10k/40k or $25\%$.
Lastly, using that $30\%$ statistics in this case is not really appropriate to begin with. All factors of production are provided by households, and that statistics is an average across both labor and capital providing households. Some capital providing households will just pay that income to another capital providing households (not even mentioning that large share of households empirically provides both). In addition, even though you specify that the 30k is only from a labor income of a households, I don’t think it is reasonable to assume that capital and labor providing households spend the same share of income on housing. The numbers for such household would likely be different from this average.