Anything which comes to your TV via your broadband connection (whether that's direct to a 'smart' TV for catch-up services, etc, or via a third-party device - such as an Amazon Fire TV stick - will contribute to your data usage). However very few broadband contracts have data limits on them these days; they're nearly all 'unlimited'. So, unless you've got a very old broadband contract with BT - which still caps your data unless you pay an additional fee - the amount of data you use shouldn't matter at all.
The only other consideration is the
speed at which that data reaches you (rather than the total amount of data used). If you're still connected to the internet entirely by copper cables (using 'ADSL'), and particularly if you're some way from your local phone exchange, then you might still have a fairly slow broadband speed, meaning that there might be problems if you were trying to use catch-up to watch Corrie in one room (through your smart TV) while your OH was watching footie in another one (using the Fire TV stick connected to a different TV).
However most people now have fibre connections bringing the internet as far as the box out in the street (and then copper cables just for the last few metres into the house), meaning that several people can all be streaming different content simultaneously without any problems.
If you want to check your broadband speed, go here:
https://www.speedtest.net/
A download speed of 10 Mbps would probably be adequate for two people streaming video content simultaneously (even if, say, they were both watching services like Netflix in HD).