Can I stay on the pump if I move to England?

Where are you coming from?
I'm definitely NOT a pump expert, but I do know that if you relocate to the UK from outside the EU, ( possibly even inside) you have to wait some time before you can have free NHS treatment. I think it's 6 months.
You can google for the information
Hana

Entitlement to NHS treatment is based upon legal residence .
Under the current Regulations, anyone who is taking up or resuming permanent residence in the UK is entitled to free National Health Service (NHS) hospital treatment in England. If your intention is to live permanently in the UK you will be exempt from hospital charges from the date of your arrival in the country but you should expect to be asked to prove your intention and that you are legally entitled to live here. This exemption applies to your spouse, civil partner and children (under the age of 16, or 19 if in further education) if they are living here with you on a permanent basis.

If you do not have an automatic right to take up permanent residence but have applied to the Home Office for leave to enter/remain on a settled basis, you will be chargeable for any hospital treatment up to the point your application is granted or until you accrue 12 months lawful residence in the UK.
http://www.dh.gov.uk/en/Healthcare/Entitlementsandcharges/OverseasVisitors/Browsable/DH_074374
The provision of a pump is entirely different . I have read of someone who had one when she arrived who continued with it in the UK.
There is also another thread with a person asking the same question here here... I'd love an update on it as I could one day be in the same position.
viewtopic.php?f=14&t=9432&p=89516&hilit=pump+moving+to#p89516
There is another thread here where a person asks the same question.