DC_Girl,
It depends on how you define "universal computer system".
If you mean a computer system accessible by any medical facility anywhere in the UK, then no.
If you mean a computer system accessible by both a private medical facility and an NHS facility, then not likely. Some private facilities may use NHS laboratories for blood work, etc., but usually the transfer of results entails sending the information out of the NHS network to the private facility, like any other transfer of information to any other third party. UK confidentiality laws governing the release of medical information are very similar to what is in place in the US, and therefore the barriers to the routine flow of medical information between NHS and non-NHS institutions are similar to the US as well. These barriers can be overcome via your provision of consent to share your medical records--just be aware that you'll likely have to go through some steps to achieve this.
If you mean a computer system accessible by both a GP and a NHS hospital within a given NHS trust, then this depends on the NHS trust. My trust has to dictate letters from the hospital (either for inpatient care or outpatient care provided at clinics based at the hospital) which are mailed to GP clinics, and vice versa. Other trusts may use the same computer system in both GP surgeries and hospitals; I don't know. You'll have to ask your GP about it.
Even if there is free sharing of information, electronically or otherwise, between all of your healthcare providers, do not assume it has been shared or looked at. The larger your file, the less likely each page will be looked at. Errors take place, information gets lost, and medical staff are busy. My advice to you is to keep the original copy of your medical records with you when you interact with healthcare professionals, and leave only copies.
How many pages are your records? Depending on the size, a one page summary may be helpful for healthcare staff, and would increase the chance that the awareness of your medical history would be taken into account regarding future healthcare decisions.