Just as a couple of people have recommended the Cabriofix - be aware the version available now (Cabriofix i-size) is different to the older Cabriofix, it's still a good seat but it doesn't last anywhere near as long and will be outgrown around 9/10 months by most babies. The older one was fantastic and often lasted until over age 2! But sadly not sold any more. Some shops drop the "i-size" in the listing, but if it has the square head support padding, it's the newer one.
Britax Babysafe Core is fairly long lasting, and so are Cybex baby seats (they have cheaper and more expensive ones). Joie are also pretty long lasting except for i-Snug. Joie Juva/i-Juva is still a good budget option, except that it doesn't fit with the majority of buggy car seat adapters. That's the cheapest thing around (£60 which isn't that cheap - but it's something). I would not touch Nania Trio, which comes up in quite a few places as a cheaper option. Terrible crash test and very outdated design. Nania Beone is OK, though - that got a good crash test result a few years ago, but the only stockist is amazon and it's barely any cheaper than the Joie, if you get the Joie from somewhere like Halfords with the offer on it.
If you get a car seat that clicks into a buggy frame (which Britax, Cybex, Maxi Cosi, Joie [except Juva] and Avionaut all fit into the multi-brand adapters, or you could get a compatible set, like a PP listed) then you could wheel this into the cab and/or unclick the car seat part and strap into a cab with seatbelt.
If you look at the Doona be aware it will only last as long as the baby is in that size car seat, which is about 18 months if you have a bigger one. Whereas another kind of buggy might last your parents 3+ years. Just to bear in mind for comparisons, because I think it ends up quite expensive unless you have a very specific lifestyle need.
Borrowing or asking around friends and family for an outgrown baby seat that they no longer need is a good idea also, just check that they still have all the parts like the newborn insert, as these often go missing and must not be replaced with an alternative. You should also try to get hold of the manual which they might still have, or you can usually download online - check it's for the exact same model, as many have similar names. Don't use a car seat which has missing, replaced or damaged parts, assume every single part is safety critical even if it doesn't seem like it is. Never use one which has been in a crash, even a minor crash. And don't use one which is more than about ten years old - they get hidden wear and tear. Other than this though, a used or borrowed seat is fine if you trust the source 