Sorry Bert, it's not clear to me - is the problem with downloading the gpx from CT or with uploading it to Strava?

Assuming you are using the CT website if the problem is in downloading from CT have a read of this: https://cycle.travel/advice/map/gps

Transferring a route manually

You can also download a ‘GPX track’ file, which can be read by all brands of GPS and many apps. This will show the route as a line on your GPS screen. Click Download GPX track, and the file will be downloaded to your computer or phone. 

Also:

Different types of GPS file

There are lots of different formats of GPS file. You can access these by clicking More download options. GPX tracks are simplest, but you can also choose a ‘TCX course’ which includes turn-by-turn prompts. cycle.travel offers these formats:

• GPX track: This is just the line of your route – no instructions. Practically everything will understand this.
• GPX route: This is an icon for each turn on your route (but no line). It’s good for GPS devices without much memory.
• TCX course: This has both the line and the turns. It’s best for Garmin Edge and Wahoo units.
• TCX activity: This is like a TCX course and is easier to import into Garmin’s Connect software.

When it comes to uploading to Strava, have a read here: https://support.strava.com/hc/en-us/articles/206811950-Uploading-Route-Files#:~:text=You%20can%20create%20a%20Strava,use%20to%20create%20a%20route.

I believe Strava will also accept TCX files. 

It may be that trying the different forms of gpx file (or TCX) will solve your problem. It may also be an issue that there is a difference between the process on the Strava App and the Strava website. 

I record my rides in Strava but use various apps for navigation. The problem I've had is that the file export is of the route without timestamps, so Strava rejects it as it cannot figure out the speed.

I haven't figured out how to export with timestamps which is a shame as there is a lot to like about the navigation on CT. 

Bert/Jon,

Like Hobbes, I am not clear what you are trying to do.

The GPX file generated by cycle.travel is for a route you are planning to do. As such, it does not contain timestamps, because you have not done it yet. I have put an example below.

A ride you have recorded and completed is usually in a non-readable format called a FIT file - Garmin's Flexible and Interoperable data transfer Format - which does include timestamps, location, speed, elevation, heart rate, cadence, power, etc, depending on the device. This type of file can be imported into Strava, automatically at the end of a ride if you so choose.

GPX files are plain text files that you can view using Notepad. In the example below, the route or track segment is a series of trackpoints with latitude, longitude and elevation, but nothing else.

Chris