Streaming and downloading have different technical requirements. When you stream from iPlayer, you are playing the programme as it is delivered to your iPad (this is a simplistic explanation). Video data are not really stored on your iPad, playing the video so you can watch it is handled by the iPlayer application (this takes up processing power too so an older iPad may struggle). On the other hand, if you chose to download an iPlayer programme, the video data is transferred to your iPad and stored there. It is then 'played' from your iPad.
So, a streamed video requires higher bandwidth because it needs to be played as it is recieved but, unlike downloaded videos, does not take up much storage on the iPad. If your broadband speeds are too low for streaming to your iPad, you could download your TV programme. Downloading may stop and start according to available bandwidth but the entire video should get downloaded and stored eventually.
You have a trade-off, higher broadband speed requirement or storage space on your iPad. Videos take up a lot of storage but you can delete them after viewing, freeing up the space again. It may be worth trying to download your programmes from iPlayer to watch after downloading has completed. However, if you don't have enough free storage space on the iPad, it won't work. If you have poor broadband speeds, with no resolution in sight, you might compensate by having a device with more (or expandable) memory and downloading favourite TV shows, viewing, then deleting them.