David Hansen wrote:

Provided the standpipe drains away properly I would fit a tee to the
bottom of the standpipe, an elbow and some pipe, to make two
standpipes close together. As long as these are tall enough both
machines should drain together without problems.

Thanks David, I think I'll attempt what you suggest rather than
fiddling under the sink.

And as a precaution, I'll try (!) not to run the washing machine and
dishwasher at the same time. I think I can be fairly confident that
there won't be any problems since the existing standpipe drains
perfectly well.