An important detail: it shouldn't be (and as far as I know isn't) the loser who pays but the party who behaved unreasonably. For example, if you damage someone's property, offer to pay £100, but they refuse your offer and instead sue you and win .... £100, then you won't be paying their legal costs. In fact, they may have to pay your costs because they were being unreasonable in bringing the case to court after you had made a reasonable offer out of court.
Unreasonableness will have an impact as to whether costs are awarded and/or how much will be awarded but the general principle is still that the losing side pays. Obviously, that will vary depending on the circumstances of the case but most cases don't encounter costs objections on reasonableness grounds.
I trhink it's more complicated than that (IANAL). Isn't it most common to make such a settlement offer "without prejudice"? Then the fact that offer was made can't be brought up in court, so it can't affect the costs ruling.
I’m not sure about England, but in the US how it works is:
The party making the offer can make a part 36 offer (or state equivalent if it exists), which grants automatic presumptions of cost sharing if rejected and the case ends with a more favorable (to the offering party) settlement or reward.
They can do a less formal without prejudice save as to costs offer, which is one the courts can use to decide cost charging, but does not get automatic presumptions of how the sharing should work.
Lastly they can make just plain without prejudice settlement offers. Which won’t factor into costs at all.
Which type of offer to make depends on who (claimant or defendant) is making the offer, and how certain they are about the amount that will be awarded at trial.
For example if a defendant is making an offer that they are absolutely convinced is more than will be awarded at trial (with the idea being that it would cost less than doing the trial without any offer, and having the normal each party pays own costs rule apply), it makes sense to make it a part 36 offer.
However when making a lowball offer, or a reasonable one but that could well be less than what gets awarded at trial, a without prejudice offer is probably better.
On the claimants side, when making a offer that they are convinced is below what they will actually win it may make sense to make it a Part 36 offer. If making an offer closer to or above what they will probably actually win, a without prejudice offer could be more sensible.
A term often used is actually "Without prejudice, save as to costs".
The meaning they're going for is "by making this offer I'm not admitting guilt, but when we come to awarding costs the fact that I tried to end this without going to court should count in my favour".
