There's a lot of confusing information around about NHS numbers in England and what they do and do not contain. I finally found something that explicitly states that the first 9 digits are randomly generated and the 10th digit is generated based on these other 9. There is no personal information in this number, including gender. As such, if you get your gender marker changed on the NHS without getting a a new number, staff won't be able to infer that you changed your marker based on that number.

NHS Numbers are randomly generated and do not include any patient information. The NHS Number has a 10 numeric digit format. The first nine digits are the identifier and the tenth is a check digit used to confirm the number’s validity)

https://webarchive.nationalarchives.gov.uk/20130503132423/http://www.connectingforhealth.nhs.uk/systemsandservices/nhsnumber/staff/guidance/factsheet.pdf

There might be other reasons to get a whole new NHS number, but this is not one of them.

This does differ by region. CHI numbers in Scotland do encode gender:

The COMMUNITY HEALTH INDEX NUMBER is ten numeric digits in length, and is constructed as follows:

• PERSON BIRTH DATE (in format DDMMYY);
• two further digits;
• a ninth digit which is always an even number for females and an odd number for males;
• an arithmetical check digit.

https://www.datadictionary.nhs.uk/data_dictionary/attributes/c/com/community_health_index_number_de.asp?shownav=0

(Also, for further proof that English NHS numbers are genderless, note the equivalent NHS data dictionary page for NHS numbers only mentions the check number

https://www.datadictionary.nhs.uk/data_dictionary/attributes/n/nhs/nhs_number_de.asp?shownav=1?query="NHS number"&rank=87.49999&shownav=1)