The required DataFrame format for a sns.heatmapyou are looking for is a pivot table with an index, columns and values passed as Date, Country and Tweet_Count to pivot, respectively, below (if this causes an error, please upgrade to the latest version of pandas as pivot had issues in past versions):
Country UK
Date
2020-03-01 400
2020-03-02 1000
2020-03-03 100
So, you simply need to pass Country to pivot as well:
uk = pd.DataFrame({'Country': {0: 'UK', 1: 'UK', 2: 'UK'},
'Date': {0: '2020-03-01', 1: '2020-03-02', 2: '2020-03-03'},
'Tweet_Count': {0: 100, 1: 200, 2: 300}})
uk['Date'] = pd.to_datetime(uk['Date']).dt.date
uk = uk.pivot("Date", "Country", 'Tweet_Count')
ax = sns.heatmap(uk)

Scroll down to the fourth block of code here:
It's worth mentioning that you can also annotate with:
ax = sns.heatmap(uk, annot=True, fmt="d")
