Hello,
Thank you both for your replies.
John, I'm not very local, so your Bedfordshire Archives suggestion sounds like a very good option, thank you for that.
Jan, that looks like it most likely is the Eliza I am looking for. There aren't many George Hine's in the area and especially not married to an Eliza. The year is a little later than I expected (the censuses consistently make her a couple of years older, as does her marriage record). However, George and Eliza only got married in October 1849, and they had a son together around 1850 (he's on the 1851 census), and I see that there's a baptism for him the same day as the Eliza one that you found. Maybe they wanted to wait until they were married before they had her baptised. They haven't got a daughter named Eliza on the 1851 census (the Eliza Hine Pollard I've been looking for appears to have been raised with her materal grandparents). So it does look promising.
I'd still be interested to know who was named as the father on Eliza Hine Pollard's marriage record, but if no one has access to it then I will think about visiting the area myself sometime, I now know of a few records that I want to look up.
Thanks for all of your help,
Matthew