He then looks at the pipework leading in and out of the boiler,
and tells me that it is actually the wrong size (15 mil instead
of a larger size- maybe 22mil?)

IIRC, 15mm is good for around 6kW, which will just about run a flat or a
well insulated small house. Should the official requirement be greater than
15mm, it won't just stop working, but may get a little noisy, as you'll have
to turn the pump speed up to compensate for the losses. This causes the
water to rush faster and may disturb the smooth flow of the water, causing
greater amount of noise than may be desiried. Provided that all the
radiators heat up well, even in the depths of winter, and the noise isn't
too bad, there is no real need to go repiping.

would I have to have a condensing boiler fitted rather than another
combi?

Not yet, although they are usually recommended. Also, as condensing boilers
are typically designed for a greater temperature difference between flow and
return, they can shift more power into the same size pipe runs, so may
alleviate the pipe sizing issues, particularly if the radiators are
oversized, allowing a 70/50 system, instead of the conventional 82/70.

You may be getting confused about the term "combi". The "combiness" of a
boiler is unrelated to its condensing nature. You would need to replace with
a condensing combi boiler. However, as you are moving out soon, you may find
it financially more beneficial to pick some cheap bottom of the range
traditional combi and hope it lasts until you move.

Christian.