I'm just about to book an economy class oneworld explorer, with the following itinerary:

LHR-MEX-(surface)-TGU-MIA-BOG-LIM-(surface)-SCL-IPC-SCL-AKL-MEL-PER-AYQ-SYD-CGK-(surface)-BKK-LHR

Flights are split between BA, AA, QF & LA.

Which frequent flyer programme would it be best to join? I'm a UK resident if that makes any difference.

Thanks in advance for any help you can give me.

Each plan is different so you will have to check each one. Remember that the redemption rate varies (a lot) between plans, so a mile is not a mile (for example AAdvantage mile is generally worth 2x QFF mile, but not always), so optimization is even more complex. Generally AAdvantage is the best plan, to get you started Earning on AAdvantage for L class is:

QF 50% miles, 100% status
LA 100% miles, 100% status
BA 25% miles, 100% status
CX 0% miles, 0 status

So you can see there is quite a bit of a difference. The CX plan (Asiamiles) might be good for a single L RTW ticket accrual, but it all depends on the split. But don't expect any FF plan to earn much for this type of ticket, it is much less than for A/D earning (making the FF plan choice not so important, after all).

Summary of my own RTW plans:
* Start in LHR because of cheap fare
* I want to start with AA Platinum challenge, therefore going westward.
* First flights are on AA only, to get 10000 miles ASAP
* After challenge there are 3 airlines that credit 200% miles to AA for class L: AA, LA, AY. All others credit 70% or less!
* Therefore my complete itinerary is:
LHR-LAX on AA
6 hops in USA on AA
JFK-HKG on AA (CX codeshare, but counts as AA)
3 hops in Asia on Japan (70% only) (HKG-NRT-SIN-NRT ?)
NRT-HEL on AY
HEL-LIS-HEL on AY for maximizing miles
HEL-MAD on AY
MAD-LHR on LA codeshare -- unfortunately I cannot book this in online tool

Total mileage credited will be 67000+
If I had the challenge done, it would be approx. 10000 miles more.

And all of this has taxes less than 300 GBP, because I started with AA.

Some Finnair Plus L class calculations:
BA 50% status and award miles.
AA 100%
QF 100%
LA/4M/XL/LP 50%
(CX/KA 100 %)
(MA 100%)
(AY 50 %)
(JL/JO/EG 50 %)
(RJ 0 %)
(IB 100 %)

Using webflyer.com the result would be minimum 38270 and maximum 47460 (if both SCL-AKL and BKK-LHR would be flown by QF). Intra-European Y class award ticket costs 40 000 and Finnair Plus Silver status would require 60 000 points (at least four segments by AY) including lounge access when flying AY. Upgrade to C to India (offer/even ridiculously cheap tourist class tickets) 20 000 OW/40000 points R/T.

Disclaimer: written after enjoying two glasses of excellent wine

P.S: Finnair counts kilometers instead of miles.