The MK-III would be somwhat outclassed by the Me-262. It has such a low powerload and overall performance. The handling advantage of the Meteor-III is somehow questionable, also. Galland flew a Meteor in Argentinia and he was impressed by it´s acceleration (FMK IV or later) but disappointed by it´s handling compared to the Me-262.
The Meteor MK-IV with Dervent V engines on the other hand is a much better contender: better acceleration, better powerload in the end better level speed. The Me-262 A could still disappear while entering a dive but all in all I see some advance for the Meteor MK-IV. In the timeframe of the MK-IV (introducing in very late 1945) we would also see some advances for the Me-262 but this is going off topic.
To the engines I also want to underline that you have to look closer WHAT engines are meaned in the Me-262.
The Jumo-004A engines for example, build with a higher degree of spare metals has an average lifetime of 50 hours. It´s quantity is neglectable (something between 10 and 15 Me-262 have been equipped with these jet engines).
The Jumo-004B 1/B-2 are prone to be very sensitive against rapid throttle setting changes, they also have only 820, resp. 840 Kp thrust output. Average lifetime is around 10 hours. A majority of the Me-262 till dec. 44 have been produced with these engines. Just to counter some generalizations.
The Jumo-004 B-3 jet engine has a higher thrust setting by some slightly increasing in distribution of additional air to the turbine blades (allowing a higher turbine temp.), resulting in an increase in power output to 890 Kp static thrust. These engine are also sensitive to trhottle setting change and have a average lifetime of 20 hours. After 20 hours the engines are taken back for excamination, if they are okey, they can be operated for additional 10 hours. Most 1945 produced Me-262 have -004B-3/B-4 jet engines. The Jumo-004 B-4 jet engines are more simplified for serial production but based on the B-3 with same performances and shortcomings.
The Jumo-004 D-0/-1 jet engines have an increased air intake and are made by new, more heat resistant alloys, increasing both, lifetime and thrust output. While still sensitive to rapid throttle settings, flame outs by slipping maneuvres are not that common, thrust is increased to 940 Kp and lifetime to + 50 hours. The -004 D was tested by some Ar-234 and Me-262 prototypes and went into serial production at feb. 1945, it is not known if , beside of a few prototypes, serial Me-262 have ever been equipped with -004D jet engines.