It largely depends on information not given so far, including the length and configuration (bends, fittings, etc.) on the 10mm section. Replacing, for example, 10% of a run of 10mm tubing with 15mm tubing is not going to make a useful difference. Replacing 90% might do. As a rule of thumb, one can transfer up to 4kW via a 10mm tubing. A double panel 1 m x 1 m radiator can dissipate about 2kW.
It's not too difficult to do the calculations, given enough details, but if you prefer practice to theory and making a mistake is potentially costly an experiment might be worth considering. Something along the following lines perhaps:
The existing 1m x 1m radiator seems to have fairly similar power rating to your new one so open its TRV fully and check and adjust the flow and return tempatures in the usual way, say 65/55 (i.e. deltaT = 10°C). Now unhook the rad and temporarily reconnect it with 10mm using lengths equivalent to the final setup (include adjustments for fittings and bends). Check that you can still get acceptable F/R temps (i.e. deltaT = 10°C). If you can't get good F/R temps you'll need to use 15mm tube. If you can, but only just, it's a bit marginal. If opening the lockshield allows you to get F/R of say 65/60 (i.e. deltaT much less than 10°C) then you shouldn't have a problem.
Obviously, the closer you can make your experiment to the final setup the better, maybe use the actual new heater for the test, but it won't have to be perfect for you to get a good idea as to whether you'll have a problem or not.