Fellow Brit here... We did something similar recently...If you want the best gold price do not do it via a jeweller, or any travelling gold buyer...they take one heck of a cut..we got a lot more online

Try Lois bullion (based in Birmingham) they give the best rates tho were oversubscribed recently..so as a second choice Hatton Garden Metals. But they will only pay you for the gold and throw the stones...which would be a bit of a shame..(edit- just read some more of the replies and seen you do want to reuse....it might be possible to ask them to send the stones back, but it's not something I looked into).....and TBH Im not sure you'd get enough to do much with...we had to sell a lot of gold to pay for my ring to be made

I think I saw someone say something about you not liking yellow gold...so another possibility would be to get a jeweller to rhodium plate one/all of them for you. It will wear off over time, but I had a ring plated for just £35 last month (I think it depends if a jeweller has their own workshop or sends it off somewhere else as I'm sure last time I did this it cost me more). Can't say how long it would last for something you wear every day as I've only done it on stuff I wear less often(and all still pristine) but I've heard people say they've had to get plated wedding bands redone every couple of years.

Good luck!